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like a Big Damn Movie but not exclusively in the realm of movies?
Yeah. It is not a parody as much as it is some type of story that takes itself "seriously".
"We are just like Irregular Data. And that applies to you too, Ri CO. And as for you, Player... your job is to correct Irregular Data."
so like, what you mean is essentially a story that takes place in a comedic setting but treats itself earnestly despite that, and never points out that, for example, the main characters are Funny Animals. i don't think we have an exact trope for this, but something like Cerebus Syndrome or Actionized Sequel is the closest i can think of
What do you call a lighthearted work that does take itself somewhat "seriously"? "Serious" in the context where it may not take itself seriously but it treats itself as a serious adventure. Think Donkey Kong Country (SNES) trilogy or the Sonic Adventure games.