I had the same issues with getting therapy and my parents not believing in mental health issues. To note, I am in the US, 18 is legal adult here, health insurance is horrible.
I personally had to wait until college to get free counseling provided by the university. They will not tell your parents if you seek help, but you have to be enrolled as a student. High schools also offer limited counseling services, but they may contact your parents if you say anything "concerning." You could go to an outside facility, but if you are on your parent's health insurance, they will see the copays. It might be worth checking out local support groups for the time being. Some may tell your parents, some may not; you should be able to ask. A full time job will often offer you health insurance. You can also enroll in your own plan if you get a job (if you don't go to college right away or aren't an enrolled college student). Sliding scale therapists exist if you don't have health insurance.
Counselors CAN commit you to a hospital, but usually only if you're at high risk of danger to self or others. You CAN deny hospital admission as a legal adult. Your counselor can also refuse to see you if you do because of refusing treatment. Whether you should is a separate question, and I'd say it depends. They likely won't try that anyway though, unless you are at a dangerous weight or have suicidal intentions.
I'd suggest reading about APA counselor confidentiality and patient rights if you are in the US.
Feel free to contact
PM/
VM if you want to discuss any of this further. I am also queer without an accepting family if you want to talk about that too. Hope this helps.