Oh my, you've dropped into my area of expertise. But before I suggest a thousand different acts, I need to know what you like about the artist you suggested. (BTW, I'd never heard Minchin, and that first song nearly made me fall over.)
My first thoughts:
If you like piano accompaniment and complex lyrics, Tom Lehrer (a Harvard/MIT mathematics professor from the 50s/60s era).
If you like the humor to be in the music and about music, like in the first number, PDQ Bach.
If it's boobies? Well, there's no shortage of songs...
I also get a feeling of 3 Dead Trolls in a Baggie:
I also can highly suggest the podcast-delivered music from The FuMP. You can also browse through the old tracks there.
Welcome To The FuMP - The Funny Music Project!