Hmmm. Interesting question.
As a Christian parent, I want nothing more than for my children to embrace the true and living God and make their own profession of faith as the Lord works in their hearts.
My own duty is to nuture and admonish them in the Lord. I don't have the expectation that my children will always agree 100% with my theological convictions. My beloved wife may not always agree 100% either. I may not agree 100% with the church officers that are over me. But submission in the Lord is required, although as the head of the family, I must respect the consciences of my family members as well. Every one of us has a duty to stand before the Lord alone as well as within the covenant relationships of which we are a part. There is a tension there sometimes due to the nature of man, but we are accountable in all of our relationships (ie., with God and man) to obey in the Lord (meaning there may be occasions when Christian children may be required to disobey their Christian parents), and that is all a parent can expect. Parents ought to be humble and not provoke their children to wrath, even as they instruct their children according to the light that is given them. Children need teaching because they are just learning about the world. But parents must teach with humility because God sometimes opens the mouths of babes to the necessary discomfort and profitable instruction of their elders.
My own prayer is that my children would be grounded in the Word of God and equipped to make their own decisions and come to their own understanding about spiritual truths.