I don't know where you go to school but I don't think the counselling center will tell your parents anything. If they deem you not able to do your work/a "threat" to costudents or yourself in the sense you disturb others by being so sad/can't concentrate on your work, they might tell the school and the school send you home for a semester, tell your parents "why" but the counselling center won't share details I don't thing.

I would go to the counselling center; your professor would be put in an uncomfortable position and would probably suggest you go to the counselling center; when I had a problem in college my girlfriend was working for the head of the Department of Health Education and took me to go see/talk to him (and stayed with me while I did, I had to promise her I wouldn't cry :-) and he was good to talk to but just forwarded me to the counselling center.

You are beyond going to parent-like figures for help, I think it would only jeopardize/ change your relationship with this professor. Keep it vague with her, tell her you are feeling depressed but don't go into the self harm thoughts, express your fears the counselling center will tell your parents (and see if she has more/better information about that) and just ask her advice what to do next.