Song parodies are such an easy form of humor that I almost think of them as cheating. Yet people continue to be impressed by the most meager efforts. The compliments seem of the "you colored within the lines!" nature to me. It may be there is a type of mind for which this just flows naturally, but not for most people.
I can do it, and do it well, but I know others who are better. I did one with a pal in 8th grade to "Ballad of the Green Beret." I bow in admiration to whatsisname, Weird Al Yankovich, who finds combinations I would not. I find it uncomfortable when people do it poorly, but even a mediocre effort can be fun. It is often only a single line that is funny, a reworking of the title or chorus, the remainder being filler. They are best when done quickly - it is too much to ask for a person to sing even a single verse impromptu (though I have heard people produce them within the song if only others will go enough verses to give them time. A rare talent.) - but dashing one off in an hour before the party is fresh enough.
We used to do them for cast parties for musicals, when I was in college. I imagine other theater companies did them as well.
I gave my beaver a four-loaf cleaver
The one he's been asking for.
I wasn't thinking, the fool that I am
He cut his tail off while building a dam.
It's no use in crying, 'cause now he's dying
And won't cleave a loaf no more.
So don't give your beaver a four-loaf cleaver
'Til you get his tail insured.