Movies that parody twenty different movies, like Disaster/Epic Movie/Scary Movie/Meet the Spartans etc.: Yes.
Movies that keep a single theme, like Austin Powers, Blazing Saddles, Space Balls, and so on: no
I agree with your premise, but Spaceballs didn't just parody Star Wars/Trek. There was a Wizard of Oz parody in there (the group approaching Yogurt), Indiana Jones (Lonestar's outfit), Transformers (Spaceball 1 into Megamaid) and a handfull of others I can't think of right now.
But that's nowhere near as broad as the modern parody movies are and I too think that that is a major reason these movies suck now but were so funny in the 70's and 80's.