I remember a send-up of Star Trek set to tunes from Hair.

Aquarius:
And you can do it with a crew that's dispensible!
Crew that's dispensible! Dispensible!
Minor characters we bring on
Perish when they meet a Klingon
... (that's all I can remember now)

I also remember a spoof of 2001 called 201: A Space Idiocy. I don't remember any songs from that, or if it did have songs.

I too have the LP "Mad Twists Rock 'n' Roll" -- it's keen, man. Although none of the songs are exactly parodies of any single song, they definitely send up various rock-n-roll song types of the day including the "my boyfriend died" genre and the "the little girl has grown up and is sexy now" genre, as well as a bona fide (and not too shabby) Elvis send-up. If that comes out on CD, definitely grab it.