No, he doesn't "help" with the baby, he's just doing what he should be doing, as the joint parent of your child. The child is as much his responsibility as yours. You need to change your mind frame about that. A man isn't a good dad just because he sometimes changes the baby's nappy, or plays with it. To be a good dad, he also needs to be a good partner, and respect his partner. Lying is not a sign of respect. Stopping his partner from seeing her family, because of his wants, ignoring her needs, is also a lack of respect.
Are you saving for a house deposit, staying at his parents? If so, is the savings account in both your names? You need to make sure it is.
I'm not expecting you to leave him right now, but I do think you need to have a good think about your relationship. It would also be a good idea to do the Freedom Programme, to learn about, and understand the signs of abuse. Many women don't think they're in an abusive relationship, because their partner can be really nice at times, and has never hit them, but abuse is complex, and there are many forms of it. Every woman should educate herself about it, so she knows if a relationship is heading that way, and can support friends or family who may be suffering it.
Your relationship has some abuse red flags, hence my concern. So, keep your eyes open, and remember what I and others have said.