Are there any other “parody” songs such as these?

Peter, Paul, and Mary did a remarkable job of mimicking The Mama’s And The Papa’s, The Beatles, and Donovan in this classic.

This clip from Walk Hard: The Dewey Cox Story is embarrassingly perfectly capturing Bob Dylan.

Most songs from Flight of the Conchords are comedic comments on the styles of others’ music. See for examples Inner City Pressure , Fashion is Danger , and Hiphopopotamus vs. Rhymenoceros .

Sid Vicious doing My Way

Spirit in the Sky by Bauhaus

Monty Python has some funny music. Sorry that I don’t have a link.

Eric Idle wrote ” the bright side of life”, in an under a hour. It would be amazing if he wrote it without basing it from another song.

Check out Mylo Hatzenbuhler on YouTube.
https://m.youtube.com/watch?v=3MjStQjCbbo
Also, this:
https://apnews.com/article/bd14585b161df8ac7e6028fa6dd85520

The Regrettes album “Feel Your Feelings Fool” is a bit of a dreamy homage to 50s and 60s doo-wap, wrapped in a riot grrl exterior. Don’t know if it’s parody or homage, but “A Living Human Girl” is funny as hell.

…And in the classical vane (pun * intentional!*) there’s anything abd everything by PDQ Bach, the alter-ego of Peter Schickele

@RedDeerGuy2 Here’s a link to Monty Python’s Always Look on the Bright Side of Life