I had the exact same problem, I was about 19 when I needed a diagnoses and the NHS took so long. The hospital they sent me to was hours away and the whole experience was horrible. The doctor made me feel incredibly uncomfortable and actually made me cry! Rather than finishing the appointment I had to leave as I was in such a state. This may have been a personal thing but I really just felt like the ‘vibe’ was off. I went private, it was quick and reliable. Yes, it is expensive but my doctor made a huge effort to minimise the amount of sessions to reduce cost. I stick to private now but obviously it is expensive. But my therapist does say that it is possible to get the prescriptions on the NHS after going private.
I would give it a few months with the NHS, obviously there mental heath department is terrible but it is worth it if you manage to get it all for free. If you really cannot cope then go private, explain you are a student and money is limited and I am sure they will be kind.
In regards to your parents, just tell them that you really need help. Luckily my mum understood but I have had many people doubt me. At the end of the day, you have to do what is best for yourself.
Hope this helps.