Stan Freberg produced many such songs in his heyday, including "That's My Boy" (spoofing sappy, sentimental drivel such as "Watching Scotty Grow"); and "Tele-Vee-Shun" (similar to Tom Lehrer's "Pollution" song). Many of his releases were songs-within-comedy sketches, such as "Wunnerful, Wunnerful!" (spoofing the Lawrence Welk champagne-music genre); "Elderly Man River" where a censor continually revises Freberg's rendition of "Old Man River" for political correctness; and "Incident at Los Voraces" (a take-off of Woody Guthrie's "Deportees: Plane Wreck at Los Gatos") simultaneously satirizing casinos, tourist attractions and the Cold War... and too many others to describe here. Do a Google search on Freberg and the first hit will be a comprehensive discography (check out the Wikipedia entry too).

Here's a link to a section of one of Freberg's recordings wherein he presents "Folk Songs for Our Time", a send-up of Stephen Foster: Freberg Underground! Show #1

There's also his spoof of Broadway musicals, "Stan Freberg Presents the United States of America, Volume 1 (The Early Years) and Volume 2 (The Middle Years)", full of sometimes-gentle and sometimes-biting satire about the nation's history and current issues.

- - - - - - - -
Garrison Keillor sometimes presents the same sort of satirical send-up songs (pale imitations of Freberg's, IMO) on his Prairie Home Companion programs.

- - - - - - - -
Big-bandleader Spike Jones was legendary for sending up his own big band! Not only did he wear loud suits, engage in physical comedy with band members, and insert numerous sound effects (cap guns, bicycle horns, etc.) into the musical scores, but he also released many songs spoofing various genres and styles. There are easily more than 100 songs to his credit.