Maybe a stretch as it’s not parodying a song but parodies Star Wars kinda sorta? Seagulls! (Stop it Now)

Lonely Island has a few (Jack Sparrow) comes to mind but again not a direct parody of a specific song.

Richard Cheese does straight up covers but they are so stylistically different that it’s basically a parody. Chop Suey

Then there’s the classic comedy music duos: Garfunkel & Oates (Loophole), Flight of the Conchords (Business Time)

Bo Burnham’s whole Inside special is fantastic too.

Looks like I’m mostly listing comedy music and not necessarily parodies, but oh well.