Do cheaters change?

About 5 months ago, I found out my current boyfriend had cheated. I am 29 weeks pregnant. I decided to stay. It’s really, really hard to trust him again. I will give him credit because he has been trying really hard no matter what mean things I have said or done, he hasn’t backed down. He got a new job as a correctional officer to make more money and provide benefits for our son when he’s born, we moved into an apartment that he will be paying for in full after I have the baby, he’s given me all his passwords, deleted Facebook and all his old accounts and emails, he’s been doing a lot. He even let me attach my email/number to the old accounts so I would’ve known if he tried to reactivate them before permanent deletion. He’s trying, but for me it’s still hard.. I don’t know if I can ever get past what he did. He comes from a great family and his mom cried when she found out what he did. My parents were upset, but actually told me to give him another chance and believe he won’t mess up again. They believe that there are bad men that don’t care about their actions and my first sons father was one of those men, but they believe my current boyfriend made a horrible mistake that he won’t repeat. Why is it that everyone can think those things, but I just can’t move forward? I want to, but it’s hard even if I can see the effort he’s been putting in... I just don’t know what to do anymore. Sorry for the long post... not sure if y’all made it this far. I just need some advice.

Edit: I should add he also wants to go to counseling (that he offered to pay for as well), but we haven’t had the time yet since he was at training for 8 weeks. Maybe that would help, but I have no idea. 😔