Tom Lehrer has been mentioned. To be specific, Bright College Days spoofs nostalgic Alma Mater songs I Hold Your Hand in Mine spoofs love songs, so does She's My Gal, so does Wiener Schnitzel Waltz, so does When You Are Old and Gray Folk Song Army spoofs folk songs Clementine spoofs about half-a-dozen genres The Elements spoofs Gilbert & Sullivan patter songs The Wild West is Where I Want to Be spoofs romantic songs of the American West Fight Fiercely, Harvard spoofs college fight songs Etc., etc. Basically, if you want genre spoofs, Tom's your man.
Somebody - Albert Brooks? Martin Mull? - did a good blues spoof, something like, Well I got up in the afternoon, I saw both cars were gone (x2) I felt so lowdown mean and dirty, I threw my drink across the lawn
Springtime for Hitler - spoofs the big musical comedy production number
Protest Song by the Rutles - spoofs, um, protest songs