Your JSA should be backdated to the day when you applied for it and will be paid until the day before you start work. I'm assuming you haven't built up National Insurance credits through work, so it will be income-related, which will qualify you for free prescriptions. You can also apply for a refund for your last prescription if it was issued after you first applied for JSA. If you're under 25, JSA won't be much, but will be enough to pay for prescriptions. "Low income" is considered to be somewhere about £15,500 (can't remember exactly). If you haven't worked since April of this year, you will pay very little income tax (if any). You should get yourself a pre-paid prescription certificate. It'll cost you a bit to start with, but worth it, because you can then be prescribed any number of medications and the certificate will cover it. I'm surprised your GP didn't explain this, but you might find that the receptionists know more about it.
I can understand why you don't want to talk to your parents, but they will notice how you are anyway and possibly just think you're lazy and a loser, so it doesn't make much difference whether you tell them or not. The relationship you have with them obviously isn't good and I seriously think you need some counselling about the feelings you have of not living up to their expectations of you. It might be that you never will (depending what their expectations are), so it's important that you start finding the real you and start being successful for yourself.
I write as somebody who was a disappointment to her own parents many years ago and now has a 21 year old daughter, who is accusing me of being the same. I'm not going to go into details, but it's a combination of these two issues which has been the main trigger for my current depression. I soooo wish my daughter, my mother and I could all talk openly, because none of us are bad people, but there needs to be more honesty and mutual respect. Before he died, my father realised that relationships and inner happiness are more important than the external trappings of success - in my case, sport - but it was too late and it makes me sad that I lost out on what could have been a good relationship with him. (Sorry! I realise that your situation is probably very different.)