Did my parents really think this was blackmail, or were they just being stupid?

That's just... stupid. In my opinion, they called blackmail on you because that's all they could do. Okay, so what if you guys were to actually break up. Then the blame would've been on them, because you pretty much made it clear that it'd be their fault, and god forbid anything be their fault. So, natuarally, it has to be someone else's, no matter how ridiculous the reason.