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I would like to apply for South Korean tourist visas for my parents, who hold Indonesian passports. They are coming to Australia next week to visit me and my sister and then flying to South Korea together. I tried to call the South Korean Embassy to check whether they can apply for South Korean visas in Australia. They said they can't because they are only holding a tourist visa in Australia.

My parents have been coming in and out to Australia in the last five years and been staying for at least three months every year. However the Embassy still seems pretty tight with its regulation. I read there is a "visa entry by designation", which seems like it is possible for my parents to travel to South Korea if they are flying to and from Australia.

Do you know how else can we approach this matter?

Is it possible for my parents to travel to South Korea if they are
flying to and from Australia?

Yes all nationals with a visa from US, Canada, Australia and New Zealand can enter South Korea with the following conditions .Ordinary Indonesia passport holders require a visa to enter into South Korea, but there is an exemption applicable to all nationals provided "Persons holding a visa (re-entry permit, permanent residency, etc.) to enter the U.S, Canada, Australia, or New Zealand (hereinafter referred to as "the four advanced countries")".

Source: Embassy Of Korea

Transit tourists bound for another country (UPDATED)

Applicable to all countries (save for Macedonia, Cuba, Syria, Sudan and Iran) that are not granted visa-free entry into Korea.

Applicable to…
Persons holding a visa (re-entry permit, permanent residency, etc.) to enter the U.S. (including Guam and Saipan), Canada, Australia, or New Zealand (hereinafter referred to as "the four advanced countries")

1) who transit through Korea to go to one of the four advanced countries. (i.e Singapore-Korea-US)

2) who stay in one of the four advanced countries and take a direct flight from the country to Korea to go to a country of origin or a third country. (i.e US-Korea-Singapore)

if you hold an e-visa to Australia and a visa label is attached to your passport, you are permitted to enter Korea without a visa. However, if the label is not on your passport, you are granted no-visa entry only when you depart from Australia and transit through Korea.

Requirements

Must have a confirmed onward flight ticket for departure within 30 days after entering Korea, in addition to having no record of criminal offence in the five countries mentioned above.

Since you have mentioned that your parents are traveling to and from Australia to South Korea means they can enter South Korea without a visa for 30 days on Australian visa.

Further Timatic confirms the same:

/ 11JAN17 / 0448 UTC

Korea (Rep.) (KR)

Visa Exemptions:

• 
  

  Passengers with a visa issued by Australia, Canada, New
  Zealand or USA to nationals of any country except for China

  
  
  

  (People's Rep.), Cuba, Iran, Macedonia (FYROM), Sudan and
  Syria, only if in transit through Korea (Rep.) must:
  -hold confirmed onward tickets on flights departing within

  
  
  

  30 days; when
  -arriving from a third country and departing to the country

  
  
  

  that issued the visa (e.g. DEL-ICN-YVR); or
  -arriving on a direct flight from the country that issued
  the visa (a visa that expired on departure from that country
  is accepted) and departing to a third country (e.g. YVR-ICN-DEL).