I'm glad you feel that you would like to work on stopping SH, that's a really big step in the right direction. Good for you!
Anyway, I'll do my best with the questions:
1) Do you feel more comfortable talking with your other parent? If so, it would probably be easier to discuss the SH with that other parent first, then he/she can help you with discussing SH with the parent who experiences the bad stress. Or you could go to your counselor at school first, and then the counselor could have your parents come in for a small group session and help explain it to them. That's more like what I did, except my therapist wasn't through my school - I was already seeing someone for depression and anxiety before I started SI'ing.
2) I still recommend the counselor way here. A counselor can explain it to your parents in a way that would be far less likely to make them freak out about it. Also, I don't know what your parents are like, but they're much less likely to get angry with you if a counselor can explain it to them, because remember: SH is not your fault - it's not a crime, but a coping mechanism even if it's not the healthiest one. And I think most counselors would really be able to help make the conversation easier than it would be with just you and your parents. (At least this is what happened from my own experience.)
3) The school counselor is most likely free. I too am Christian and I understand you not wanting to talk to a pastor or someone from Church about this. It's definitely easier to talk about SH to someone you don't know and you won't have to worry about judging you later. Plus, if you're close with any of those people from the Church, you might also fear hurting them or hurting your parents who know them. But that's the great thing about a counselor - they won't judge you and they won't be hurt by you. They'll only try to help you.
4) If you're not a legal adult (18 in the U.S.), I think they might be obligated to tell your parents, but I'm not really sure as I didn't start SH'ing until I was already an adult. But like I said above, a counselor at school might be the best way
to tell your parents!
Best of luck with everything. You're very strong to be trying to stop and to want to ask for help. I am
so proud of you for doing this. Please let me know if there's any other way I can help. (I'm on here whenever college isn't devouring my life, haha.)
*Hugs* <3