I know you feel desperate to get started with your transition; I doubt there's anyone here who doesn't know how that feels & who wouldn't sympathise with your situation. But if you go behind your parents' backs and use their policy - that they pay for - without their permission for something you know they're vehemently against, you run the risk of betraying your parents' trust and permanently damaging your relationship. The question is: is that risk worth it? If it saves your life, then I'd say it is definitely worth it. But think carefully about the long-term consequences: how are they likely to react if they find out?
But that's a problem for a later date. Right now, you need to take care of yourself and that probably means getting started on your transition. Your parents have made the decision to keep you on their policy and pay for whatever medical care you need. That's their decision, but that's where their influence ends. Because you are an adult, they have no right to interfere with the type of medical care you choose to receive, even if they've decided to pay for it.
As an adult, you do not need your parents' permission for anything, even if you're still living under their roof because you're living under their roof as an adult (like a house guest or a lodger), not as a child. It sounds like your parents are still treating you like a child: if you want them to start treating you like an adult, you need to drive that forward yourself by demonstrating your independence. Nobody ever gives you permission to do this; you have to do it yourself. That's very hard to do when your parents have been overbearing and you'll meet resistance at first, but it has to happen if you're to take charge of your life.
A fully-grown woman may do as she pleases; if it pleases you to go to the doctor, then that is precisely what you should do. And if you do so, remind your doctor that under State law you are entitled to strict confidentiality, which means your parents are not to be contacted or informed about the purpose of your treatment, and you could also request that all correspondence must go to a different address (e.g. college, or a friend's house) to protect your confidentiality.