"Parents are their children's most important teachers and mentors, and they bear primary responsibility for nurturing their sons and daughters while keeping them safe. Yet parenting is one of the most undervalued and least prepared for roles in America." (emphasis added)
Being responsible means being answerable for the thing with which you are charged. That parents bear primary responsibility for the nurture and health of the children means that they are answerable for using the tangible and intangible resources at their disposal to nurture their children. The two most critical factors for the healthy development of a child are love and language. In the presence of appropriate parental affection, and the respectful, loving use of words, children develop normally (unless there is an underlying disability, in which case love and language are even more important).
Most Undervalued Role in America
Parents play the most important role in any society. Period. They have greater influence over the character, competence, creativity and health of their children than anyone else. Failing to recognize this fact is to undervalue the role of parents, and as Edelman rightly notes, parenting is the most undervalued role in America.
That has two consequences. First, it means that we as a society don't hold parents responsible. We hold schools responsible for the behavior and education of children rather than those who 'bear the primary responsibility.' Second, it means that we don't support parents in their critical role. (Happily, there are excellent exceptions to this generalization, including Parents as Teachers.) Instead, we look for ways to shift the responsibility of parents to programs and institutions. Yet however well a program can perform, it cannot replace a child's most important relationship: the relationship with her parents.
Moving toward responsibility
Parents bear the primary responsibility for the nurture and health of their children. The decisions that we make have far-reaching consequences for the character, competence, creativity and health of our children. To fulfill that responsibility, we need wisdom. We need good information, and we need to use it wisely. If we are wise, we will not only use that information not only for ourselves, but for other parents, which is precisely how the network of support and accountability can and should function. And finally, we need to be prepared to give an answer for our parenting. Apart from this, the word 'responsibility' has no meaning.