For someone over 18, there are a few pieces of legislation involved in this (I'm studying mental health nursing so we have to know all about this). The first is the Data Protection Act, which says that legally, so long as you have capacity and there are no safeguarding issues, then what you say goes when it comes to your data, so the GP and any other services you might be referred to cannot tell anybody anything about you without your express permission, even if they were to phone the surgery and directly ask whether you were seeing them for a mental health problem.
The second is the Mental Capacity Act, the main principles of which are: Everyone is assumed to have capacity until proven otherwise; capacity is defined as the ability to understand information about a decision, hold that information long enough to make a decision about it and communicate that decision; capacity is decision specific (so someone with dementia might have capacity to decide what to have for breakfast but not where they should live for example); and people have the right to make objectively poor decisions - it doesn't mean they don't have capacity. If you were somehow deemed not to have capacity to decide to keep this from your parents (which I think is incredibly unlikely), then the principle of best interests can be applied, and your parents could be told without your permission. But like I said that's incredibly unlikely. Perhaps if you were so depressed that you couldn't bring yourself to speak - but I guess by that point your parents would probably have realised something wasn't right.
The third is the Care Act, which outlines laws on safeguarding - essentially if you said something that made the GP think you or someone else were at risk from someone else (or someone else was at risk from you) then they have a legal obligation to report that to the relevant authorities. But unless your condition was relevant (for example if you had a child but your depression was making it difficult to care for them properly) then your condition would not be disclosed.
Finally, there is the Mental Health Act. Again, it is very rare that this gets used in cases of depression, but if you had suicidal plans and were unable to be supported in the community, then it might be deemed necessary for you to have a stay in hospital. If you were opposed to this then the GP might contact some colleagues to assess you to see whether you could be detained under the mental health act or "sectioned". But again, even if this happened, they wouldn't tell your parents unless you didn't have capacity and they thought it was in your best interests to tell them. I would hope that if it got to this point, however, you would tell your parents and not just disappear on them.
Having said all that, sometimes services do slip up. They wouldn't directly tell your parents anything, even if you were a danger to yourself (though they might strongly encourage you to), but they might send you a letter that your parents might see and question you about, or phone and leave a message for you to contact the service that your parents might hear. But to get around that you could make up something as an excuse for what the letter is about and you could make sure the GP only has your mobile number. And if you make it really clear that you don't want your parents to know, they can put an alert in your notes for people to be careful when contacting you.
To make an appointment, all you need to do is phone up the reception and ask for one. They usually ask what it's about just so that the doctor can be prepared, but you can just say that you'd rather not say & they won't press you. Having said that, it might be a good idea to ask whether any of the doctors have a particular interest in mental health and make an appointment with them, because not all doctors are that great with mental health, so seeking out one that does will give you a better chance of being taken seriously. In the actual appointment, I'd go in with a clear idea of what you're going to say. A simple sentence to introduce the problem, like "I think I might be depressed" or "my mood has been really low recently", and then what you are hoping for from them, whether that's medication, a referral for therapy or counselling. They will probably ask lots of questions and they might want to do a blood test to check that it doesn't have a physical cause.
Anyway, sorry for the essay - I hope some of it helped! Whatever happens, remember that GPs see this kind of thing all the time. Depression and anxiety are so so common, GPs are used to talking about it and generally know the right questions to ask. They aren't going to judge you or think you're weird. Good luck!