A patriarchal society is one in which the male head of a family has a great deal of power. In such a society, the father of the family can pretty much do as he wants and he can order the other members of the family around. They are expected to do as he says. To us, this seems like a pretty old-fashioned way of doing things.
Romeo and (especially) Juliet do not fit in to such a society because they (especially her) are willing to go against their fathers' wishes. We don't really know much about what Romeo's father thinks, but we do see that Juliet's father is really angry at her. He is ordering her to marry Paris (like a patriarchal father would do) and she is disobeying. She continues to disobey him even to the point of being willing to kill herself rather than marry Paris.
So the point is that these kids (especially Juliet) are not obeying their fathers the way they are supposed to in a patriarchal society.