I’ll bet most people on the Board won’t recognize Courageous Cat and Minute Mouse, which was a made-for-TV cartoon series from 1960. They live in the Cat Cave, travel in the Catmobile and the Catplane, and can be called by the Cat Signal searchlight. Instead of a Utility Belt with lots of gadgets, he has an apparently inexhaustable supply of themed “guns”. It’s very obviously a parody of Batman and Robin. What’s surprising is that it’s by Batman creator Bob Kane (plus Gerald Rappoport)
In 1967-8 Marvel comics had Not Brand Ecch, which ran parodies of their own superheroes (and DC ones, as well), written and drawn by the usual Marvel staff, often the very ones who wrote and drew the original characters. It ran 13 issues
After writing The Hobbit, Tolkien wrote Farmer Giles of Ham (although it wasn’t published for another dozen years). It’s been argued that this was a sort-of parody of The Hobbit and adventure tales in general, with Giles a reluctant hero who sort-of defeats a dragon with the aid of a named sword. There are a lot of scholarly in-jokes.
I have seen a lot of one-panel or one-page joke and parody strips by comic book artists (there’s a great drawing by one of the Superman artists showing Superman, carrying Clark Kent’s clothes, angrily banging on the door of a phone booth so he can get inside and change, but Alfred E. Neuman refuses to get off the phone), but I’m looking for a long story or sustained series.