In some cases, a birth parent can indeed stop an adoption from happening. Whether or not it is possible for a birth parent to stop an adoption from happening depends on a great many factors, including whether the parental rights have been revoked by a court, whether the adoption has been finalized, whether the baby has been born, and whether legal paternity has been established.
If a woman agrees to voluntarily give up her child for adoption, for example, she can stop the adoption from happening right up until and for at least a short time after the baby is born. The amount of time that she will have to change her mind will vary, but it may be as little as a few days, or as long as a couple of months or more. You should be certain that you are aware of what the regulations in your state are in terms of the birth mom changing her mind, as the time can vary greatly. Some insurance companies even offer adoption insurance to potential adoptive parents, so that they can recoup some of their expenses if this occurs.
For a birth father to stop an adoption can be more difficult. First of all, not all birth fathers are even aware that they are birth fathers. In addition, if a woman is pregnant but not married, her word is legally binding as to who the father of the child is. If the birth mother claims not to know, then no one can legally be considered the birth father. The exception to this case is when the woman is married. If she is married, her husband is legally the child’s father. It certainly is possible for a birth father to establish paternity with testing and such, but often this process can take longer than the pregnancy, birth, and adoption process.
If the birth parents have had their rights revoked in a court, they cannot stop an adoption from happening. At that point, they don’t have any rights over the child whatsoever. To stop an adoption from happening, a birth parent would need to fight the court’s wish to take away her parental rights, and would have to keep from voluntarily relinquishing those rights.