It has been proved by DNA that he is the father, so he does have parental rights.
Makes no difference whatsoever!!!
You aren't married. He therefore isn't even ALLOWED, legally, to register the baby. It's entirely your decision.
GIVE THE BABY YOUR SURNAME.
These boards are full, FULL of women who are no longer with their shits of exes and in many cases aren't even getting maintenance, and they bitterly regret giving their child the surname of a family and a man who doesn't give a shit about them.
Please don't give in. It's absolutely NONE of his business, which is why he's trying to bully you - he knows it's your decision and he has NO RIGHTS here at all.
He's already causing issues for you. Do you really think he's likely to be around in two, three years time?
And just think, why the fuck should your baby carry his name when he's not going to be doing the hard yards?
Your surname. Your baby will live with you, be in your family, let her have the security of the same name as her mother.