I can't really give much exact advice, because health service costs (or lack of) change per country, people's parents are different, etc.
Assuming it's not pointlessly expensive, I'd recommend going to your doctor alone and explaining things to them. In this country, at least, they may be able to refer you to a psychologist/psychiatrist (can be free, at least over here). They may not necessarily be much help, and I'd heavily recommend against starting on any medications unless you cannot even begin to understand and remove what's causing you problems, but doing such and then informing your parents would, in many situations I believe, be enough to make them begin to realize how serious you are about this. From there, you could begin (again, depending on country) to get into a welfare system for the time being.
I was in a pretty similar situation when I dropped out of university. The thing I went there to learn ended up not being something I was really interested in enough to go through several more years of study, tests and a heavily flawed education system. So I just stopped going to classes and wandered around the campus instead, then eventually dropped out. It all went a lot easier than you'd think. From there, if you mend things with family (you don't have to be doing this for them or anything, it's quite fine to see it as for your own peace and comfortable living conditions) and take a break from stressful life, you'll gradually become more open to doing things and may find something you'd be happy to study or do for a job. That is, on the condition that you keep an open mind, and don't develop aversions to those things, which it seems many of the longer-term NEETs tend to. Basically, enjoy the NEETlife, but don't feel anger, resentment or depression for the "outside" world. That was what held me back for the longest. If that sounds or becomes something that's difficult for you to do, meditate every day. I'd be happy to give advice on this, if you'd like. It's easily the single most effective path towards an open mind, willpower, control of your emotions, and the complete destruction of suffering. I learned that a breakdown, if handled well, is a chance to purge what you've learned is harmful for you and completely start over.
Finding a community of online friends can be helpful to some, but finding satisfaction in solitude is also helpful and without the clinging and trust and other emotional risks that come from depending oPost too long. Click here to view the full text.