I would also add Walk Hard: The Dewey Cox Story and Popstar: Never Stop Never Stopping as great newer parodies. The first, starring John C. Reilly is a spoof of Walk the Line and music biopics in general, the second a spoof of Justin Bieber: Never Say Never and popstar documentaries in general, starring Andy Samberg. Both movies were box office bombs unfortunately, despite being really funny and also having actually great songs.
Airplane! is also a parody of one particular movie that is mostly forgotten now, Zero Hour, and spoofs a then popular subgenre of airplane catastrophe movies.
Walk Hard in particular is very much in the ZAZ vein.