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Korea does a covid test on arrival and holds you in the airport until clear.
I believe this is just to be allowed in the plane in the first place.
Total travel time is going to be about 15 hours.Doing the PCR test on Sunday morning should be fine for all the potentially times, except very nervous that the test result won't turn up on Monday. Which doesn't give much time to get another test.
I'd assume the test has to be no more than 72 hours old at the point your dad lands in Korea, that is the case for some parts of the world.