No advice, but I live with my parents too so I just wanted to let you know you're not the only one. I am saving and saving and saving, and one day I will get my own place. That keeps me going.
How long have you been with your parents?
tbh though, my parents babysit for me and help with collecting and dropping off and so on. Also, luckily this house is big and we have enough space for all of us, and it works - ish although I do long to fix up my own place. It'd be lovely, but it'd be hard too.
I'm on the council housing list too, but I don't expect to be a priority. But I've been on it for 2 years now, so I suppose eventually, eventually I'll hear something positive back.
Make the most of it while you can. Use your parents to babysit. Get out some times. Save a little tiny bit of your lone parent allowance if you can.
Because you live with your M&D, could you work in the evenings/? your son would be at home asleep, so it wouldn't cost you childcare expenses?
I know it's hard, but it's a means to an end, and one that with a small child you can put to your advantage in some ways.