Doctor's are bound by confidentiality, so I don't think they're allowed to tell your parents anything about your health without your say so.
Here's an excerpt from the General Medical Council guidelines:
As for not feeling anxious about your appointment, I think a lot comes down to the doctor you visit. I've seen patients with pretty serious conditions laughing away because of the personality of their doctor.
42. Respecting patient confidentiality is an essential part of good care; this applies when the patient is a child or young person as well as when the patient is an adult. Without the trust that confidentiality brings, children and young people might not seek medical care and advice, or they might not tell you all the facts needed to provide good care.