Possibly. Some states do allow a minor to get therapy without a parent's involvement. What is the reason that you don't want your parent involved in this? As for discussing suicidal thoughts, I have done so with my T however, some people here have had bad experiences. A therapist is required to report if you seem like you are really going to commit suicide. Otherwise, a T can keep it to themselves. It is hard to know when to report and when to not report. I suspect that having a parent involved might make the T more comfortable with the arrangement because a parent can help keep you safe if it comes to that.

I would enquire with a trusted adult. Perhaps ask a teacher, guidance counselor, pastor or someone else. I hope you get the help you need.