I've successfully fought off copyright BS (I do song parodies) but ill tell you this: it's not worth dancing the line. If you do a song parody, make the music slightly different (enough to obviously not be the song) than the real song. Don't give them the opportunity.
They take it, even if they are not in the right. Companies abuse legal channels as a business tactic (parodies can sometimes become what people remember of a song and they don't like that) and there is nothing you can do to change it.
It sounds like you weren't just dancing down the line though. To have it get to this point demonstrates a lack of understanding of what's going on, unfortunately. I don't really know how to fix it when it gets to this point, just prevent it. Claims I've fought off have never been on the border of killing my channel.
Also, they really take a lot out of me. I've devised ways to clearly not be infringing rather than knowing that legally I am not infringing but having to defend it.