First of all, you are blaming yourself for things I'm 99% sure are not your fault. Your parent's marriage did *not* get messed up because of your ED! Perhaps the fact that you were going through something exacerbated a problem they already had (disagreements about how to raise you, communication problems, etc) but there is no way a healthy marriage would have been "messed up" by their daughter developing a psychological issue! If they can't deal with little bumps along the way with a child they are raising, there are deeper issues that you have no control over.
Secondly, how close to 18 are you, and how bad is the anxiety?
I'm pretty sure it is illegal for them to tell your parents details about what you have said to them, but when it comes to a minor being issued medication, they might have to check with parents. And I'm not sure how the health insurance system works here, but when you are under 18, isn't your insurance technically bound to your parents? In The States, it's like that, so they would need to be notified about any medications I was receiving, simply on the basis of doctors trips, insurance bills, and actually obtaining the medication.
Further, a lot of doctors really don't give much credence to the idea of privacy for minors! When I was under 18, I had way too many experiences with doctors and therapists betraying my trust. You agree to give them an inch, and next thing you know, 5 people have been notified of your "problem" in detail. Happened too many times before I was 18. Maybe you should discuss your concerns about your privacy with the doctor in question and see what he says. Just tell him your parents have reacted badly to psychological issues you've had in the past, might not buy into the "anxiety mumbo-jumbo" and you'd prefer for them not to be involved. See what he says.