JOEL - What you are talking about occaisionally happens in LDS homes, much to the dismay of the parents. And it does hurt very much if the children decide to stray from the Gospel. It means that the parents might not have a complete eternal family in Heaven. The scriptures are very direct and strong in the responsibility that is placed upon parents concerning the Gospel.
"And again, inasmuch as parents have children in Zion, or in any of her stakes which are organized, that teach them not to understand the doctrine of repentance, faith in Christ the Son of the living God, and of baptism and the gift of the Holy Ghost by the laying on of the hands, when eight years old, the sin be upon the heads of the parents."(D&C 68:25)
It is the parents responsibility to see that their children are taught and raised in the gospel, and they have every right to encourage their children to be active in the Church. And the children should honor their parents wishes when it comes to religon. But it should not have to be done with force but with gentle persuasion as we are told in our scriptures:
"No power or influence can or ought to be maintained by virtue of the priesthood, only by persuasion, by long-suffering, by gentleness and meekness, and by love unfeigned; By kindness, and pure knowledge, which shall greatly enlarge the soul without hypocrisy, and without guile-- Reproving betimes with sharpness, when moved upon by the Holy Ghost; and then showing forth afterwards an increase of love toward him whom thou hast reproved, lest he esteem thee to be his enemy;" (D&C 121:41-43)
It is not easy to make a statement on something like this that would apply to every situation. According to the policies of most religions the child would not be allowed to make an official change to a different church without the parents permission until the child is at least 18 years old. What happens in situations like this depends on the type of relationship the parents have had with their child. A lot of prayer, attention, patience, and love are needed to encourage the child along the right path. After the child has reached an adult age they can decide for themselves what religion they want to belong to.