Garfunkel & Oates (Riki Lindhorne & Kate Micucci)
Uncle Bonsai/Electric Bonsai Band/Mel Cooleys (Andrew Ratshin &o.)

A number of comics do musical parodies and write original songs, but, understandably, you'll find them more on the comedy stages than at music venues.

Martin Mull did a lot of funny songs before moving more into acting. Check out his bottleneck ukulele on YouTube.