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Firstly, good on you for being concerned about her and asking about it on a public forum where others who may not be able to can hopefully benefit from this as well. I hope you come back with your findings from the airlines/trips they take!

Basically, it comes down to the airline. You can see what their policy is by looking up their Conditions of Carriage. Very few do specify exactly something about weight/height/size, but will merely talk about 'a passenger's physical state causing discomfort to other passengers or crew', which is more likely to be used against those intoxicated, or if you tried to board while say, covered in manure??

A friend just came back on a flight from Honololu to Auckland with Air New Zealand (see related query on Travel.SE), and the person next to them was physically unable to fit into a chair with the armrest down. As a result, all three in the row were forced to endure an awkward and uncomfortable 12 hour flight. Air New Zealand's conditions of carriage only mention 'physical state', but at the time we looked around and found airlines like Southwest in the US are starting to actually mention size/weight in their conditions of carriage.

Your best bet is probably to look up the conditions of carriage, and if it's a little ambiguous, tweet or email the airline and ask them directly. If you have a statement from them in writing saying that there's no problem, that's a very easy document to have your folks print out and take along, just in case there's a problem. If there's not, no embarrassment, it just stays in their bag and they get to tell you there was never anything to worry about ;)

EDIT

An update, I had a flight a few months back with Qantas, where I was wedged between two very large women. The one on my left was big enough that the food tray couldn't be lowered, and she couldn't locate her own headphone jack in her seat. I found at the end of the flight her similar-sized husband was a row back, because the two of them physically couldn't sit next to each other on the same flight.

It was, as you might imagine, uncomfortable. I spoke with Qantas afterwards about their policies, which they were a bit evasive about, but tried the line "we can't tell how big passengers are until they arrive at checkin". I pointed out that the same applies to my baggage, but they have a size limit on that :/

Their current policy seems to be that if they can, they'll move the inconvenienced passengers, but if it's a full flight, tough luck, and IF you complain enough, you'll get a token appreciation in frequent flyer miles, which is a shame.

I am 6'3" and pushing 400 pounds and I have never had a problem on BA flights. Yes I need a seatbelt extension but only so I will be comfortable. I admittedly do fit between the armrests.

I do feel bad for people next to me because my shoulders are quite broad but I try to get an aisle seat so I can at least lean out.

Since your mother is travelling with your father it should be fine. They will even be able to put he armrest up if she needs more space.

Not sure if this is useful, but based on experience I would say that someone the size of your mother will be fine.

My girlfriend is 5'4" and almost 450lbs right now, and she is just getting to the point where a second seat is truly necessary--up until recently she could squish (albeit uncomfortably) and with the courtesy of the other passengers and a single extension she could do just fine. At 300lbs she had no real problems at all. The seats were a little tight, but the armrests could still go down at that point. :)

Recently watched a show on TLC called My 600-lb Life and a lady who was 661 lbs. was shown flying on a US based flight on an unmentioned airline (although it appears to be a Southwest plane seat) and who required three seats with a belt extender. If one is of this size or smaller, it would seem that accommodations exist for one to fly.

Here is a screen shot from the show of the lady sitting in and requiring three (3) seats of the plane:

(My 600-lb Life | Erica's Story | Feb 8, 2017 | Copyright © 2017 Discovery Communications, LLC.)

I've flown Southwest for years and never had an issue until my last trip I got home from yesterday. I'm a big girl 5'6 and close to 400 ( working on it) and usually fly 1-2 times a year. I've actually lost weight since the last time I flew. I generally ask for a seatbelt extender and I'm good and I can get the arm rests down... its tight but it works. Last Saturday though I was sitting at the gate waiting for my flight when I was loudly humiliated by a Southwest supervisor telling me very loudly and classlessly that I needed an additional seat to fly with them ... they did not charge me for the extra seat but the humiliation of that moment and flight I will never recover from. It was terrible. Which is what brought me to this page. Others mentioned it but seeing if the airline has policies about it is helpful but also totally depends on your body shape too. Sounds like your mom is smaller than me and I've literally never had an issue until last week.

If you're willing to buy her an extra seat to avoid embarrassment, then why not just buy an extra seat anyway for their comfort?

I think this is less an issue of what's allowed and more an issue of what makes sense. Most domestic or intra-European flights fly planes with slim seats that make even slim seats uncomfortable.

If you can afford it, I'm sure your mom would appreciate the "business-class" seats. Or... perhaps even look at upgrading them to business class. It may be the same price as 3-tickets and they'll have much wider comfortable seats with nicer service.

I have a few larger family members so I’m extremely sensitive to situations as this. However, I fly several times a year and am constantly sitting next to someone who is too big for their seat and it’s unbelieveably uncomfortable. In fact, I’m currently on my third flight in the past 2 weeks and am currently leaning into the aisleway to make room for the other passenger. Like I said, I'm sensitive to these situations and wouldn’t dare do or say anything to embarrass the passengers, but if you can, I would buy the extra ticket. It’s probably just as uncomfortable for them. I know you wouldn’t be bothering other passengers, but your father deserves to be comfortable too.

Also, my friend is a flight attendant and she usually just hands certain passengers the belt extenders as they board to avoid the embarrassment of them having to ask later. Once a passenger needed one and was given one but didn’t use it. She kept a blanket over the lap so they didn’t notice till the end of the flight. His is a felony and was such a silly reason to get into so much trouble.

I am 5'4 and roughly 256. The seatbelt fit around me ok, but it does depend on the way your weight is distributed throughout your body. Southwest is pretty strict on making people buy a second seat if they are intruding on the person next to them. I recently flew march 10,2017 and there was a guy on the plane who had to purchase a second ticket and he was given two seats. It all depends on the way your body is shaped really. I hope this at least helped a little bit for anyone that is reading this.