The parent plays numerous roles and has numerous responsibilities. The general answer is that the parents must relay the principles of the Torah - meaning, the way to perform its laws as well as the ethical principles the Torah teaches, in short everything that is involved in becoming a Torah-observant Jewish person.
This can be gleaned from numerous Torah verses that specifically address the principle of parents teaching the children. One of them, I know, offhand, is in part of the "Shema" (Devarim 6) - "You shall teach them to your children..." Mind you, I'm sure that there is a specific commandment regarding this, whcih most likely comes from a different verse (would appreciate assistance on this, please.)
I know that my answer states the principal in general terms, without specifying HOW to accomplish this. But, this general rule, I think forms the essence of the parents role. And this responsibility is continuous throughout both the parents' and child's life. It could be argued that teaching children, in a sense, even occurs while the parent is dead, but that's for a separate question.
Responding to your comment, below:
Avot ("Ethics" of the Fathers) stresses in a few places that action is more important than scholarship. Thus, the most important role parents have is to actively show how mitzvot are performed and, more importantly, to never demonstrate to your children that the performance of any commandment is "burdensome", even if you, personally feel that way. Ideally, you shouldn't. But, for purposes of mentoring your children, I think you should, temporarily, "fake" the joy, or at least hide the burden. Your kids should always sense that every thing you perform is for the sake of God, and you perform it because you love to obey God's commands.