You’re confusing satire with parody. Parody is when you use the protected work to comment on that work, while satire is when you use it to comment on something else.
If you take R. Kelly’s “I Believe I Can Fly” and change the lyrics after the first verse to be about the singer jumping off a roof and having to spend the next three months in the ICU, that’s parody — you’re making fun of the song’s lyrics by exploring the consequences of an overly-literal interpretation. On the other hand, if you used the melody for a song with totally different lyrics about clubbing baby seals, that’d be satire.
If you’re sued over “I Believe I Can Break My Spine,” you have a good excuse for why you used the copyrighted ”I Believe I Can Fly” you can’t exactly make fun of the lyrics without using substantial copyrighted elements of the song.
On the other hand, for a satirical song about baby seals, you don’t need to use “I Believe I Can Fly.” You’re using it because it’s easier than coming up with your own melody, but if you just want to avoid extra work you need to get a license.
The distinction can get fuzzy when a work is also clearly talking about other issues.
Nader’s U.S. presidential campaign in 2000 featured this ad, which was clearly based off Mastercard’s “Priceless” ads. The ad was most directly criticizing his opponents as corrupt politicians, and that would be satire. However, when sued he argued that part of the subtext of the ad was commentary on Mastercard’s ultimate message of materialism (sure, there are “priceless” moments, but you can get them with things money does buy).
In contrast, a book in the style of Dr. Seuss about the OJ Simpson trial was held to not be parody — it didn't make any point whatsoever about Seuss, just about the trial. Seuss’s style was just used to draw attention, not because they were giving commentary on Seuss.
In any event, fair use is complicated. There are four statutory factors:
- The purpose and character of the use (specifically including whether or not it's commercial)
- The nature of the copyrighted work.
- How much of the original work was used
- The effect of the new work on the market for the old work.
All four statutory factors have to be considered; none of them can automatically make something fair use or not fair use. Even a legitimate parody might not be fair use under the right circumstances.
The only way to definitively know if something is fair use or not is if you’re actually sued for copyright infringement and convince a court that it was fair use. It’s fairly common for whoever’s backing your work to demand that you get a license even if it’d almost certainly be fair use, because they don’t want to have to go to court and argue that as a whole the four statutory factors suggest that your work was fair use.