Parental Rights in the Army?

You do need to go to your base's legal assistance office and find out where to begin (although it's going to be hard to the get the time to go down with your pending deployment and training requirements). Without that original birth certificate, you can't get the kid into DEERS; also, without the custody arrangment determined by the courts, you're in a position to get screwed; really, there's no legal reason to stop your girlfriend from taking the kid to her parents while you two are deployed; and, you can't develop your Family Action Plan because according to your service, you don't have dependents. So, right now, your girlfriend really can take your child to her parents.