No. You can if you're inclined, but usually you don't have to.
I've read that Weird Al Yankovic asks for permission from the original artists, or at the very least asks if they wouldn't mind if he parodies their music but as far as the law goes, he doesn't necessarily have to since Parody is protected.
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Generally, it is called a parody. But not all parodies are meant to make fun of the original. Some are made just for fun, and not to offend the first one.
Cinderella Parody in Sgt. Frog is episode 196 part 2.
In general, parody existing is a good thing, though there are subjectively good and bad parodies.
If the song is COMPLETELY your own - the music, then sure.
Dan the Song Parody Man was born on October 23, 1963, in Englewood, New Jersey, USA.