I guess this is a matter of opinion, but I would think it's a good gesture to invite all the people to the 1st birthday party who came to the shower, or at least the ones who sent a card or gift. It's an extra way of saying "thank you" for helping out with a new baby.
You can certainly invite additional people to the birthday, but definitely extend an invitation to anyone who helped the parents prepare for the new baby.
I hope you post back to let everyone know what you decided to do. This is a really interesting question. Thanks for sharing.