The Tom Lehrer revue Tomfoolery (which incidentally right now has 3 productions - in St Louis MO, Portland OR and DeKalb IL), has a 1-2-3 punch in the second act that can't be beat with "Smut", "Oedipus Rex" and "Masochism Tango".

And I just about fell out of bed laughing the first time I heard his "Vatican Rag" on the radio, back in 1968.

My all-time pick for funniest recording would have to be the Stan Freberg (with Daws Butler) cover of the Platters' "The Great Pretender", where the jazz pianist does not want to play that "klink klink" stuff.