Devil's advocate kind of because I can see both sides...
Your dad's advice differs from your mum's because he has never been financially dependent on her in the same way. It's easy for him to assume that you will always be happy to live with your partner and he will always be happy to live with you.
I would be alarmed that my daughter was painting herself into a corner in your situation too. Not so much with the working part time but working part time from home for your DP's dad.
It could all go as it is now which is great! But there are risks and your mum is right to point them out to you - not that you are a shit person at all, but that you are leaving yourself in a vulnerable position.
Imagine if you and your DP aren't getting on, you want to leave but you'll have no job - or at best a job working for his dad - and no marriage protection over assets. While this might not happen; it is nice to know you are in a relationship because you want to be and not because you have to be because the alternative is awful.
Same if your DP leaves you.
As I say it sounds really doomy - especially as you are loved up at the moment. But I have seen enough good relationships go bad and have had friends who've spent ages in shitty relationships where neither person is happy to caution against giving up independence completely.
So I wouldn't necessarily advise working full time as that's really hard for the baby and new mother but would suggest doing the legal marriage bit at least and once the baby is here look for a PT role with a different company!