I have never encountered a thoroughly successful systemic approach to morality, and certainly there have been a great many attempts to develop them. From realist appeals to moral sources, descriptivism, prescriptivism, emotivism, psychological necessity, down to deontic logics, almost all of them fail to capture one or more critical aspects of morality while almost all of them seem to capture some aspect or another. Even moral relativists
usually speak of moralities as being objective within a particular setting.
In my tentative view, what is most often the failure is the desire to impose objectivity in moral codes wthout considering how "universalizability" works in the real world. Universalizability is the idea that rules, in this case moral ones, appy to all equally. Theorists often attempt to incorporate (i.e. presciptively enforce) "universalizability" in either the structure or the particular content (usually both) of this or that understanding of morality. I think this is where a primary error lies. It is the unthinking drive for Truth in morality. I suspect that trying to capture Truth in morality is about as reasonable an enterprise as claiming certainty in knowledge - which is to say, not at all. (Go figure; I'm a skeptic, right?)
I suspect it is simply enough to permit for the possibility of universalizability without requiring it. Therefore, I suspect that morality really has more to do with what I like to call realms of reasonable expectations, with "realms" and "reasonable" being defined via constant negotiation. This idea is a work in progress - I'm trying to take a step forward by taking a step back. It seems to me that, like our other social contructions, morality is what we agree to think it is.
Now this provides me with an answer to anyone who asks for the source of my morality. "I derive it from the reasonable expectations of myself and those around me." How interesting this answer is an open question...
I approach the role of genetics with a degree of wariness. While there well may be something interesting to consider there, I question whether it might turn out to be another quest for absolute Truth in morality, but based on biology this time. I suspect it is the enterprise itself that is in error. Biology (genetics) may suggest tendencies, but to refer to them as morals is, I think, a misunderstanding of what morality is and how it works. But, of course, it's another idea to bring to the negotiating table.
Reflecting back on the original post, I entertain the idea that parents, in fact, do have the abilty to reinforce morality in children by teaching them the importance of respect for the reasonable expectations of others. If that primary understanding is developed, the rest, it seems to me, follows...