Weird al is a rather obvious.
But Axis of awesome is also a sheer favourite of mine. their four chords song is a genius composition mashup of many iconic songs throughout the years.
Whitney Avalon is a very underrated one who does some really good and rather witty pop culture themed songs.
Least favourite: Dennis Leary (He lives up to his name)
I know this is meant for english speakers, but for francophones folks around here, Ultra Vomit is a great band if you’re into metal and dumb jokes (like this song parodying the french Hokuto no Ken opening, talking about (Heine)ken beers).