We can't diagnose of course, as you already realize.

That said, it really doesn't sound like BP to me. Online quizzes (that I've seen anyway) are particularly bad at differentiating between bipolar disorder (BP) and borderline personality disorder (BPD). And frankly, they don't even seem to try to (mood swings are not exclusive to BP). There are significant things that differentiate BP and BPD. What you've described sounds more like Borderline and an eating disorder. Neither cutting nor starving have anything to do with bipolar.

I'd take your parents up on their offer of seeing someone. Seeing a professional is really the only way to know what might be going on.