It's a tricky one.
I think it's very sensible to decide that you don't want to date until you're stable, but then who decides that you're stable enough to date?
If you've told him about the Mental Health stuff, then that's certainly a step in the right direction. If he's fine with that, then that's good. I don't mean to make him sound bad, or anything, but he might not know exactly what to expect, so he might have more of a problem dealing with it than he thinks he will. That's the kind of thing that you can only find out by giving it a go though.
I don't think that deciding who to date by who you think your parents would approve of is the right way to go about things. I'm sure if your parents see that he makes you happy, then they will come round to him. Especially as they get to know him.
It may be that being in a relationship will be positive for your Mental Health anyway. Certainly I've found that I'm generally better when I'm in a relationship, as it gives me something happy, and something good to concentrate on
One thing I would say, though, is that if you want the relationship to work, then you need to be honest with him about how you're feeling. So if you're feeling really rubbish, then don't pretend to be ok to him. He'll be able to sense that you're not ok, and it'll lead to trust issues. It's much better to say "Yeah, I'm feeling rubbish at the moment, so I'd rather not see you" than to make excuses about why you don't want to see him.
Good luck either way
