If you formally adopt the children, then you have all the rights of a birth parent. Otherwise, I believe you have to concede to the birth parent's wishes.
Unfortunately you really don't have any rights unless the biological parent was unable to care for the child and you were awarded custody. You can make rules for your home.
You have many or most of the same rights as if you were the real parents. However you should work something out with the "real" parent to set up boundaries and understanding with the child. Stepparents get less respect then the real parents so you need to be strong.
