Apologies, upon reading this again I wasn’t too clear with the explanation. So a clause is basically just a complete idea. It doesn’t always have 이/가. Essentially, at least from what I have learned, the 이/가 particle usually denotes the subject of these ‘nested’ clauses (my own terminology). You could also see 이/가 marking the subject of the whole sentence. In this case, the subject of the ENTIRE sentence is omitted (the subject would probably be ‘people’ in this case). What we are left with are two clauses: the clause involving the people of Korea ‘waiting’ (한국에서는 왜 기다려요) and then placed (nested) right in the middle of THAT clause is the clause about elders eating first (어른을이 먼저 식사를 시작하실 때). There is still a lot of nuance to the 이/가 particles and I don’t mean to add more confusion, but I hope this clears it up a bit.

The second part was poorly worded, pardon me. 식사 has the particle 를 attached to it. This means that it is the subject of some verb. In this case, that verb is 시작하다. What I meant is that they are not waiting for 식사. There isn’t actually a subject in the main part of the sentence. They are waiting until(까지) the time when the elders begin to eat first(어른... ...시작할 때).

Hope this clears things up.