Their jokes about assigning gender to babies and to being transgender, dressing in drag, like all of it was a send-up.
Sure, they did punch down if you were a person who were in those groups, but the fact that it was large enough social event to be relevant enough to be a comedy skit on a television show or a movie seen by millions implies that there were some serious things going on back then that they could see and wanted to address.
What the hell was going on that put all of those things in their mind?
Dressing in drag had been a staple ofEnglish comedy for decades if not much, much longer In the first world war guys were doing it for concert troupes (essentially, vaudeville acts by and for the soldiers.)
It literally goes back to Shakespeare, something all Brits are completely aware of.
In addition, transgender people have always been around, and it seems as if the conservatives have been terrified of them forever. So there’s obvious potential for comedic value.
Echibit A: Queen, I Want to Break Free.
This is the most correct answer. English Theater, historically, had men playing the women’s parts. There’s a long, long tradition of it. So MP dressing in drag follwed the tradition, the viewer might find it both appropriate, and humorously anachronistic, along with MP’s exaggeration of the characters makes for great humor.