In the circles I travel in, it's called filking. One of my fellow filkers wrote a brilliant parody, which, alas, I can't remember to "Ghost Riders In the Sky" about an encounter between a spaceship and a dragon. It ends with the spaceship about to lay an egg.
"Ghost Riders" is one of those songs that begs to be filked.
I wouldn't parody "Green Fields of France". But Poul Anderson filked "Waltzing Matilda" in a song about the food at a Westercon called Bouncing Potatoes. "Was that a musketball that was fired at Lexington? No said the waitress that is a pea."
Parodying has a long and honorable history. Lots of old songs have half a dozen sets of lyrics addressing various issues of the day.
Dutch farm boys once sang a song that went "Yanke diddle dudel down". About the pay they got. When it came to this country it turned first into a funny song about colonial yokels in the army then a patriotic song about Washington's brave volunteers. Which didn't stop tories from singing a song about Benedict Arnold to the same tune. Parodying is a legitimate part of the folk tradition. It's also a hobby of mine.
So:
There is a mess in New Orleans
How did it come to pass?
'cause Dubya's man in FEMA, my friend
Was a big fat horse's.............feedbag.