This is a fun question!
If you want to make fun of the "angsty missing dad" trope, some things that come to mind off the top of my head are either a) dad is missing, but it's for a totally silly/trivial reason (for example: one day he accidentally missed his stop on the train, and after that a whole chain of other random wacky events transpired that kept him away from home for years) or b) dad is actually fine and it's all a misunderstanding (for example: dad went on vacation spontaneously, protagonist missed his message saying so and became convinced that dad must have disappeared under sinister circumstances).
Or maybe because the dads in these kinds of stories are always doing some kind of secret research, part of a secret organization, etc, you could maybe do a story where the protag thinks that his father was doing some manner of mysterious important work that the protag must now take on in his place, but (again, lol) it's actually a misunderstanding and the truth is the dad wasn't actually anyone special. (Sorry if all these ideas are too silly for what you have in mind, btw. )
These are serious examples, not parodies, but in general I think two really good examples of classic angsty missing dads where it turns out later on that there's a bit more to the story are Ging from HxH (as mentioned above), and also Hohenheim from Fullmetal Alchemist.