VH1 Classic just played "Synchronicity II" by The Police. I'd never seen it before, but instantly recognized that it was a quotation of the iconic video for "Dancing with Myself" by Billy Idol, whose personal image Sting is imitating.
Seems like it's been a long time since a well known group made a video that quoted another video. This brief review confirms the hunch. In 2011, All-Time Low made a video spoofing Katy Perry and other major stars du jour, but they were not a very popular group (never appearing on Billboard's Year-End charts). The last clear example was "All the Small Things" by Blink 182 way back in 2000, and even that was a blip, as there were no others from the '92-and-after period. (Needless to say, Chris Rock did not appear on the charts, and I don't recall seeing the video for "Champagne" back then.)
After "Synchronicity II" in '83, David Lee Roth made a more obvious and slapstick parody video for "Just a Gigolo" in '85, the same year that Phil Collins aimed for parody in the video for "Don't Lose My Number." In '91 with Genesis, he included a Michael Jackson parody in the video for "I Can't Dance." There are probably some others out there that I can't think of immediately, and that the writer of the review missed, but that's enough to establish the basic picture.
Those four acts were all major successes at the time, all appearing on the Year-End charts across multiple years, hence the parody videos would've enjoyed high visibility.
So why the decline since the '90s? Well, allusions like these rely on the audience having already been exposed to the original and stored it in memory. Steadily over the past 20-odd years, less and less of the target audience has reliably tuned into music videos -- and music in general. Moreover, music videos have gotten a lot more bland and featureless -- and who's going to remember a forgettable video when it's referred to in a later parody?
By "forgettable," I don't mean "inferior" in a subjective way, but objectively lacking in identifying features -- things that help you explain them to others to jog their memory. Simple memory tests would show how easy or difficult a video is to recall. "Y'know, that video where the glamor models are playing the instruments and the singer is wearing a shirt and tie?" (Actually, he made three videos like that -- the one where the models are wearing pink, or wearing black? And is he wearing a suit or just a shirt?) Try doing that for recent videos -- "Y'know, that one where those rappers are boasting around a pool and a bunch of big booty hoes are shaking their asses..." Oh right, that one.
I think this also explains why parody songs have died off since the heyday of Weird Al Yankovic in the '80s and early '90s. You can't get the listener to recognize the target song if it has no memorable riffs, melody, solo, or distinctive vocal delivery. It can only end up sounding like a parody of the whole genre, whose performers are all interchangeable (emo, crunk, etc.). That would also have been difficult in the '50s and early '60s, when most pop music sounded so similar and lacking in identity.
Thus, the peak of pop parody during the '80s and early '90s suggests a peak in those years for the distinctiveness of both the music and the videos of hit songs.
Related: Ornament is meant to make things more memorable