Was the house/flat you moved out of rented?
Was it rented in your name, his name, or both names?
Are you still at your parents'?
Are you on maternity leave at the moment, with a job to go back to or not?
My thinking is that you need a trial separation, at the very least. Ideally, you'd stay in the house/flat and he would find somewhere else. Unfortunately, you moving into your parents can seem a bit like you just going on a visit. That said, moving in with your parents isn't necessarily the end of the world if it means you get help, support, and can raise some money for independent living.
I think you do need to have a conversation with him, where you spell out what you and your child now need in terms of his support and behaviour. He isn't shaping up, for whatever reason, right now, and I think it might help you to make it absolutely clear to him that a relationship with you, as opposed to just hanging out with you and kvetching, requires a good deal of imput on his part. Or, frankly, it's not a relationship, is it?
Personally, I think you should think about a separation, and set a date a week, fortnight or month or so hence, where you get together and he either shows you what he has done to make a relationship plausible or you both decide you are separating - with all the shared commitment to your child that that requires, actually.
I know several men who have gone very pear-shaped with the appearance of a child. Some of them have, given the red-card, shaped up. Some haven't. some have seemed to, and then fallen apart later. I can't tell you which one of those he'll be. But I do think that you need to bring the situation you're in to an end as soon as possible - moving it to some clearer ground where you have control, rather than drifting along at the sidelines of his drama.
Obviously, if he's in a bad, traumatic place, this is very sad. It really is. But in that case he needs a friend, supporter, etc. He is not being what you, and your child, need. Unfortunate, but true.