How would I be able to ensure that the custody of my child would go to my parents rather then her father? He did sign the birth certificate and pays child support, but her father is a well known alcoholic with 2 DUIs, has no license and continues to drive under the influence. He has well known rage issues, hit me on one occasion and has put several holes in walls. I don't fear for my life, but if something were to happen I would want to make sure she grew up in a more stable environment then what he can provide.

I hate to say it, but if the father wants custody, it is very unlikely that he won't get it. Don't misunderstand, if he doesn't want custody, no one is going to show up and drop a child on his doorstep.

Your parents could contest it, but if he fights for it, 9 times out of 10 he wins.