Parents have a responsibility to teach their children gender-normative behavior for the same reasons that they ought to teach their kids good manners.
Manners, customs, and fashions serve to create a social environment in which the behavior of other people is predictable and facilitates cooperation. While many aspects of culture can vary dramatically without any social impact (ethnic foods, dance outfits), there needs to be a consensus on other aspects of optimal social function. Some manners are arbitrary whereas others have an objective basis. For example, greeting others is objectively important for conflict avoidance, whereas whether we shake hands, or touch noses is arbitrary. Yet there needs to be a consensus even on arbitrary customs when they facilitate critical social functions. You are expected to dress up for an interview or funeral, even if the definition of “dressed up” is different in Wall Street, San Fransisco, or Hawaii. Attending an interview while sloppily dressed creates uncertainty about the applicant’s future behavior: does he lack culture, respect, or work ethic?
The above is merely to say that it is important for customs and manners to be broadly in consensus in a society or community. We have to judge other people in order to understand their intentions and establish relationships. Customs provide the language by which we mediate our relationships. We can teach our children to be open-minded in the sense of being open to new ideas and different ways of life. However, they still need an inner framework to establish healthy relationships.
In the case of gender, extensive research has shown that boys and girls have biological predispositions to gender-specific behavior. Boys play with cars and girls play with dolls mainly because of biological reasons (though the exact mechanism may be unknown), not cultural ones. Other gender differences are purely cultural. Girls wear dresses and ponytails and prefer pink mostly due to historical chance. Thus, if we want girls and boys to adopt these gender-normative customs, they have to be taught. This usually happens automatically and unconsciously when parents, relatives, and friends encourage gender-normative customs.
My point is that this is a good and necessary aspect of our culture for the same reasons that teaching children to say “hello” and “thank you” is necessary. Without such direction, customs would become unpredictable, causing other people’s behavior to become unpredictable, and civilization to erode.
Men and women specifically need modalities of interaction to temper undesirable aspects of our sexual drives. Primitive human societies without such customs tend to engage in perpetual violence, both between males and in male-female relations. Animal societies which lack culture entirely are far more brutal, for example, by routinely killing or expelling young offspring when a new dominant male takes over. Culture and established gender norms were invented by man to keep such violence and chaos in check. Having said that, the vast majority of boys and girls are prewired to accept gender-normative customs, and eager to adopt them.
The above is not a comment on transgender people. It is a comment on parents who attempt to raise their children in a “gender-neutral” way. There is no such thing — biology will assert itself one way or another. All that “gender neutral” parenting accomplishes is to fail to educate their child on an important aspect of social interaction and cooperation and handicap them in their future social relationships. I am not suggesting that parents ought to raise children to behave exclusively in stereotypical gender-normative ways. I’m suggesting that parents should (1) acknowledge inherent sex differences between girls and boys and (2) prepare them for successful social and romantic relationships.