I've been living on my own since I was 16, I've struggled to find full time work as a designer, with no formal qualifications and barely enough experience to call myself a professional. I recently lost the first, well paying and only full time office based design job I've ever had. You know what? Fuck it, I don't care. And you shouldn't either! You're 20, you made a "life decision". Well done, seriously. You had the hangers to go for it and it doesn't matter if things didn't go to plan. Your parents are reacting the predictable way, they still care but it's their job to bitch and complain. Tell them that it's your prerogative to fuck up, and it's not theirs to make it worse.

You quit your job for perfectly good reasons, it was a crappy job and if I was in your situation I'd have done the same. Be proud you walked away.

You're going to be fine if you keep your shit together, use your friends taking the piss as your justification to succeed. Go find a crappy job that'll earn you a piss poor amount of money and then go focus on being awesome in your spare time. Make something random, see what happens. Repeat. Find other devs/designers to work with, see what you can do. Repeat. Work in the day and do crazy shit at night. Get your confidence back, build your portfolio. Get to a point where you've got your cahones back and then do something with yourself. Move to somewhere where there's cool companies looking to hire, impress them all. Repeat.

You'll get somewhere if you push for it. And if you fuck up? Who cares! You're young, just enjoy the ride.