While both are grammatically valid sentences they mean different things, and yet is surely what you'd want to use in this case.
Person A: Is your father here yet?
Person B: Yes, he has just arrived.
(Note the use of arrived rather than come; come needs an additional "from where", ex. "Yes, he has just come home from work.")
When using yet here, you're asking a question: is your father here. You don't know the answer, and you expect a yes or no reply.
It's a bit different when you use already:
Person A: sees or hears something that makes them realize the father is here, but they didn't expect the father to arrive until later
Person A: Is your father here already?
In this case the question is a rhetorical one; Person A is aware that the father is there, but is surprised by this fact. So they state that the father is there already, as opposed to later on. They don't expect an answer, because they already know. This is just a way of expressing surprise.