That is not a Weird Al song.
Weird Al NEVER made a song about Barney on fire.
Avril Lavigne's Complicated
"I Want it That Way" by the Backstreet Boys
its a parody of "Ridin" by Chamillionaire
Its just the music of rage against the machine in general that hes making fun of.
"Piano Man" from Billy Joel.
It's not making fun of a particular song. It's more just in the style of the 50's "Doo-wop", Elvis type sounds.
Black Gold Soul Asylum
Weird Al never does a parody of any song unless the musician who wrote that song gave him permission. So he was clearly not punished at all for that parody.
This may not be the best choice of a song for school, since there are doubtlessly some students in your school who are fat, and who may feel that you are trying to make fun of them by using that particular song. Although it is a brilliant song.
It's an original, not a parody. However, originals are often in a song style of another group. This one is like Guns 'N' Roses