RE: Do Parents Stereotype Their Children's Gender?

Gender stereotypes are adaptive mechanism. By acknowledging the inherent physical superiority of the male children, the females focus their efforts in counterbalancing their physical disadvantage via development of other means of interaction. Adult men tend to communicate differently from women, with men experiencing difficulty verbalizing emotions. Could it be that women learn and develope emotive language as a counter-force against male physical dominance? What would the modern West produce, as they forcibly reject the very fact of biological differences between the sexes? Is the rampant youth violence and "epidemic" of ADHD observed in our modern era a consequence of penalizing "boyish" activities of physical nature by predominantly women educators?

females focus their efforts in counterbalancing their physical disadvantage via development of other means of interaction.

Some of the research agrees with you. Here's a rather lengthy quote:

Serbin and colleagues found that the increase in influence attempts by girls was almost entirely an increase in making polite suggestions to others, whereas among boys the increase took the form of more use of direct demands. Furthermore, during this formative two-year period just before school entry, boys were becoming less and less responsive to polite suggestions, so that the style being progressively adopted by girls was progressively less effective with boys. Girls' influence style was effective with each other and was well adapted to interaction with teachers and other adults.
These asymmetries in influence patterns were presaged in our study with 33-month-old children: We found then that boys were unresponsive to the vocal prohibitions of female partners (in that they did not withdraw), although they would respond when a vocal prohibition was issued by a male partner. Girls were responsive to one another and to a male partner's prohibitions. Fagot (1985) also reported that boys are "reinforced,' by the reactions of male peers--in the sense that they modify their behavior following a male peer's reaction--but that their behavior appears not to be affected by a female's response.
My hypothesis is that girls find it aversive to try to interact with someone who is unresponsive and that they begin to avoid such partners. Students of power and bargaining have long been aware of the importance of reciprocity in human relations. Pruitt (1976) said, "Influence and power are omnipresent in human affairs. Indeed, groups cannot possibly function unless their members can influence one another" (p. 343). From this standpoint, it becomes clear why boys and girls have difficulty forming groups that include children of both sexes.^[9]

Written by a woman, btw, so no sex bias here!