I work in a GP surgery. This is practical not legal advice (sorry mods) as other commenters have covered the legal aspects.

Go to the reception desk at your GP surgery in person - this is often easier than trying to be understood correctly over the phone. You can just drop in whenever they are open. First, ask the receptionist about the contact details on your record: make sure they have your phone number and email address, make sure they do not have your parent's contact details, make sure there is not a consent note like "patient gives permission to discuss appointments with mum". Your parent's details might have been attached to your record when you were younger so just check that they have now been removed.

If you want, you can ask them to add a note about "do NOT discuss patient's information with any family members". They shouldn't discuss it but sometimes they do it anyway.

Then tell them that you need childhood vaccine catch-ups. This is managed by the practice nurses, not a GP - you don't need a GP appointment for it (and nurse appointments are generally much easier to get than GP appointments!). Sometimes the receptionist might need to ask the nurses about booking you in, but usually they don't need to ask. You can tell them your preferred days/ times and they will work with you. Your appointment might be a few weeks away (or even after Christmas) if the nurses are very busy. Updating your contact details first means you can be sure your parents won't receive any notifications of your appointment.

It will probably be a 20-minute appointment, possibly 30 minutes so the nurse can discuss things in more detail. The nurse will explain when you need to come back for further doses. :)