That's the definition of a song parody. I mean, that's literally what the term means. It's when you take an already-existing song and rewrite the lyrics. Trying to say it isn't a song parody is to go against what the term means. It's like trying to convince someone that ducks aren't birds; by definition they are.
You're right. But it is a parody.That isn't satire.
I don't know if I'd call it a satire. A satire involves actual criticism of something. Great examples of satire are 1984, Animal Farm, and Brave New World. That's satire.The word parody gets thrown around way too often in my opinion.
Airplane is a satire of all the disaster movies around the time.
Airplane, from what I could tell, wasn't trying to make any real point about the disaster film genre, it was just using the framework as an opportunity for jokes. That's a parody, but not satire.