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The answer is NO. There are two reasons why you cannot do this, especially on an international itinerary.

1. It is a violation of the contract one of them entered into with the airline when purchasing the ticket. The ticket is issued to one person only and cannot be transferred.


2. 
   Both the US and El Salvador require APIS (Advanced Passenger Information System) which transmit all passenger details, including travel docs and passenger demographics. The one who does not match will likely get denied boarding or, worse, detained at the entry point.

If you try to change any passenger details on the return only, meaning after they have been verified on the outbound, you risk forfeiting the ticket.

No.

This is explicitly a violation of United Airlines' Contract of Carriage. At best, you will likely have the ticket voided. At worst, you may end up violating Salvadoran and/or U.S. laws related to making false statements regarding passenger identity for purposes of flight safety and/or immigration.

Assorted selections from UA's Contract of Carriage that say you can't do this and what the UA-imposed consequences may be if you try (note that this doesn't include any possible civil or criminal consequences that might result from breaking passenger identification or immigration laws):

Rule 6 - Tickets

G. Tickets are not transferable unless otherwise stated on the Ticket at the time it was issued. The purchaser of a Ticket and/or the Passenger intending to use such Ticket is responsible for ensuring that the Ticket accurately states the Passenger’s name. Presentation of a Ticket by someone other than the ticketed Passenger renders the Ticket void, and UA is not liable to the owner of a ticket for honoring or refunding such ticket when presented by another person. If a Ticket is in fact used by an unauthorized person with or without the knowledge or consent of the person to whom the Ticket was issued, UA will not be liable for the destruction, damage, or delay of such unauthorized person’s baggage or other personal property, or for the death or injury of such unauthorized person arising from or in connection with such unauthorized use. As used herein, “unauthorized person” means any person other than the person to whom the ticket is issued and who is entitled to be transported or to a refund in accordance with the rules in this Contract of Carriage.

K. UA’s Remedies for Violation(s) of Rules - Where a Ticket is purchased and used in violation of the law, these rules or any fare rule (including Hidden Cities Ticketing, Point Beyond Ticketing, Throwaway Ticketing, or Back-to-Back Ticketing), UA, without notice to the passenger, has the right in its sole discretion to take all actions permitted by law, including but not limited to, the following:

1. 
   Invalidate the Ticket(s);


2. 
   Cancel any remaining portion of the Passenger’s itinerary;


3. Confiscate any unused Flight Coupons;


4. 
   Refuse to board the Passenger and to carry the Passenger’s baggage, unless the difference between the fare paid and the fare for transportation used is collected prior to boarding;


5. Assess the Passenger for the actual value of the Ticket which shall be the difference between the lowest fare applicable to the Passenger’s actual itinerary and the fare actually paid;


6. Delete miles in the Passenger’s frequent flyer account (UA’s MileagePlus Program), revoke the Passenger’s Elite status, if any, in the MileagePlus Program, terminate the Passenger’s participation in the MileagePlus Program, or take any other action permitted by the MileagePlus Program Rules in UA’s “MileagePlus Rules;” and


7. Take legal action with respect to the Passenger.

Rule 21 - Refusal of Transport

UA shall have the right to refuse to transport or shall have the right to remove from the aircraft at any point, any Passenger for the following reasons:

A. Breach of Contract of Carriage – Failure by Passenger to comply with the Rules of the Contract of Carriage.

E. Proof of Identity - Whenever a Passenger refuses on request to produce identification satisfactory to UA or who presents a Ticket to board and whose identification does not match the name on the Ticket. UA shall have the right, but shall not be obligated, to require identification of persons purchasing tickets and/or presenting a ticket(s) for the purpose of boarding the aircraft.

G. Across International Boundaries – Whenever a Passenger is traveling across any international boundary if:

1.The government required travel documents of such Passenger appear not to be in order according to UA's reasonable belief; or

2.Such Passenger’s embarkation from, transit through, or entry into any country from, through, or to which such Passenger desires transportation would be unlawful or denied for any reason.

Source: United Airlines Contract of Carriage

Aside from the fact that this violates the Contract of Carriage, some answers have mentioned that you could change the date of birth for the reservation online after the outbound flight, but before the return flight. Note that this may not be possible with U.S.-based carriers, such as United. While it appears to have been possible on British Airways, as another answer posted, I checked this with Delta and it does not work. Instead, you get this message:

Source: Screenshot from Delta.com

The message says (emphasis from the original):

Your flight is less than 72 hours away. We cannot update Secure Flight Passenger Data within 72 hours of your flight. If you need to make any updates, please see an airport agent.

Note that I was not within 72 hours of the return flight when the screenshot was taken. The outbound flight had been completed several hours before the screenshot was taken and the return flight was still more than 72 hours away. However, the Secure Flight Passenger Data fields remained locked out after 72 hours prior to the outbound flight, so no changes were possible online between the outbound and return flights. While I haven't tried this with United, I suspect they will be more similar to Delta than to British Airways in this regard, as both are U.S.-flagged airlines.

The answer is: yes. it's possible. When your father is flying out of the USA, make sure you update the Date of Birth and Passport number in the Passenger information section of Online Check-in. Be aware that some airlines lock out online changes 72 hours in advance, so its best to try as soon as your first leg lands. Remember that technically your two flights are separate, even though both were purchased as part of a single booking.

If that's not possible for some reason, simply visit the check-in desk before the flight and ask to have those details amended. If asked why the original information was wrong, simply claim that you have no idea how it happened. The worst penalty for this small trick is that your ticket is forfeited, but the chances of getting caught are minuscule.

Finally remember that the airline doesn't really check details such as passport number and birth date, so you might as well skip the check-in desk and head straight for boarding without amending the APIS data. However be aware that this may result in your father's departure getting recorded incorrectly, as the US doesn't use exit controls.

Example BA APIS update screen after taking an outbound flight to the US:

My answer is "Yes", but there is some explanation due. It looks like some honorable SE fellows mixed together different issues. So let's address those separately. Please keep in mind I'm not a lawyer, so if you need legal advice, please contact one licensed in your jurisdiction.

1. Would you be breaking the law? I don't see any way how you could. As long as correct travel documents are presented (i.e. you present your passport when you check in, your father presents his passport when he checks in), no immigration or passenger identification laws are violated. It is up to airline to transmit the proper information to authorities, and if their system doesn't allow changing a passport number, it is their problem (but this would be plain dumb because a dual citizen can have several passports under the same name).



2. 
   

   Would you be breaking United contract of carriage, and what could happen in this case? Theoretically you might be breaking the contract of carriage. However most likely nothing would happen. You did not enter DOB when you purchased the ticket, only the name. At check-in desk you're presenting a valid, confirmed ticket, and a valid travel document, which matches the name of the ticket, and represents the person. As long as you're not making any statements (such as stating you were the person flying the other leg), I see no valid reason to deny you boarding.

   
   
   

   Please note that it is up to the court to decide whether contract breach really occurred, and how exactly the other party should be made whole. There is no guarantee the court would agree with United (especially if jury trial is chosen), especially considering that United is not suffering any actual damage in this case (only "missing profit", but this is hard to prove), and considering the nature of the contract (there were no negotiations). The chance is about the same as getting sued for using "hidden city" booking, or throwing away the return leg on a round-trip ticket - both are violations of Contract of Carriage, but to the best of my knowledge nobody ever got sued for doing that.

   
   



3. Is it a good idea?. It depends on the price difference between round-trip and two one way tickets; in some markets the difference may be minor, and you will be essentially paying for a peace of mind. So it is up to you whether the (minor) risk is worth the price.




4. Is such restriction in the contract even legal? This is an interesting question. As far as I know, the typical airline conditions of carriage were never tested in court. It is obvious the parties are not equal here, and the contract is incredibly one-sided, there is no meeting-in-minds. Would be an interesting case.

Regarding the contract of carriage: the United contract of carriage only allows United to refuse boarding if the name on the ticket does not match:

Rule 21 Refusal of Transport

E. Proof of Identity – Whenever a Passenger refuses on request to
produce identification satisfactory to UA or who presents a Ticket
to board and whose identification does not match the name on the
Ticket. UA shall have the right, but shall not be obligated, to
require identification of persons purchasing tickets and/or presenting
a ticket(s) for the purpose of boarding the aircraft.

Nowhere else the contract says the DOB must match, only that it is collected by United as "mandatory government information". Thus I fail to see how he could be refused boarding.

Because we're talking about international travel, you'd risk the possibility that someone in border patrol could consider this to be fraudulent use of an identifying document for the purposes of entry into the United States, which carries a penalty of up to 5 years in prison. A decent lawyer would probably get dad found not guilty - at trial, three months after he was was arrested.

There seems to be some disagreement / ambiguity in the question. reirab showed that this would be a violation of the ticket purchase agreement, so it's definitely not unlawful. It could go very wrong, depending on the opinion and attitude of whichever Customs and Border Protection Agent happens to get the call.

Again, I'm not saying dad would be guilty of a federal crime (though he might be), it's certainly entirely possible that one guy working for border patrol would think this fraud qualifies, and you winning at trial doesn't mean you actually win if you've been sitting in jail awaiting trial.

Let's try to do analyze it step by step, ok?

An airline company's duty is to bring someone from point A to point B. As obvious as it seems, we must remember that they are required to just check the match "person-valid ticket-valid travel document", everything else (e.g. immigration) is not of their concern.

So, what's the matter with names on tickets and not being able to change them? Because the airline companies to stay profitable need a great amount of control on ticket prices, thus what they do is to remove the chance of resellers by making tickets non transferable.

Which lenght they go to reach that goal is left to every company internal's decisions. There are companies that requires you to insert your travel document number in their systems when booking, for example, while many others (most?) just make do with the name on the ticket. I mean, seriously, can you imagine a scalper trying to make money by buying tickets with random name hoping to selling to other people with the same name? :-D

So far, so good: if your airline just ask your name and nothing else, your plane seems doable.

But then, there are at least two problems. Rememember: even if you purchased the ticket with just first name and last name, planes' tickets are still not transferable and so we need to do a distinction here between legality and feasibility before going on.

What you are planning to do is illegal; no matter how many loophole you plan to use, you are still doing something illegal.

Now that this is perfectly clear (and I strongly believe you already knew this before asking), let's go on to analyze the two problems I was saying before.

1. The less obvious: when you board at the gate your passport or ID card will be scanned and registered. Now the airline company which didn't ask for your travel document number while booking or online check-in has your data, has them matched with your departure ticket, and as the return ticket is bound to the departure's one, they have your data to check when you'll be back. Ooops.




2. Immigration. When you fly away from your country this information and your passport data are registered. Sooner or later (and I'd say sooner) some bell will ring in Immigration offices pointing out that someone entered (legally or not, it doesn't matter) in the country by exploiting a namesake...

Conclusions

Is it legal from the airline company point of view? Not with 99% of the airlines companies.

Is it immoral? That's your call, not mine.

Will you be caught? You can bet on it.

Should you risk it? Again, your call.