How does Romeo and Juliet portray patriarchal society?

Verona at the time of Romeo and Juliet is shown to be a patriarchal society in many ways. Generally, we see a contrast between the freedom of the male characters and the seclusion of women. Romeo and his friends have the freedom to stay out all night and crash parties without consequence. They fight with the young Capulets and face legal consequences for disturbing the peace, and they are shown to suffer little restraint. Romeo enjoys the reputation of a lover, smitten with Rosaline before meeting Juliet. He is free to pursue his romantic interests without censure from family or society. In short, the young Montagues and Capulets are typical rich kids with wealth, freedom, and no responsibility. It's that very combination of factors that drives the feud between the two families, as the patriarchs themselves seem to have little inclination to break the peace.

Juliet is the only young female character, but she clearly leads a very different life. She spends most of her time within the Capulet family compound, accompanied at all times by her nurse. She can go out to go to confession—the excuse she uses to marry Romeo in secret—but otherwise, she is kept under close watch.

Juliet's principal value to her family is as a pawn to form an alliance through marriage. As such, she is cloistered, lest she damage that value through romantic adventure. Her father plans to marry her to Paris, kinsman to the prince, and in this transaction, we see the full extent of how patriarchal Verona society is. First, her father arranges the marriage after Juliet meets Paris just once, and he does so with no input from his daughter. Second, he makes the decision without consulting his wife. Finally, when Juliet refuses, he threatens to disown her.

The role of a patriarchal society in Romeo and Juliet is largely shown through Juliet's relationship with her father, Lord Capulet. Juliet is encouraged to be subservient and obedient (like her mother, Lady Capulet) and to marry the most desirable suitable, Paris, despite the fact that she is not even fourteen years old. The systematic arranged engagement of young women at this time was part of a strategic effort to create mutual benefits between two families. In this case, Paris is a nobleman and kinsman to the Prince; he is handsome, rich, and offers significant financial value and social status for the Capulets. Juliet is a well-mannered and beautiful girl who will make a dutiful and attractive wife for Paris.  Such a union would strengthen both families, even though it would come at a high emotional cost to Juliet, who is too young to even have considered marriage. 

When Juliet later begs her father not to make her marry Paris, he responds with great fury. As the patriarch of the family, he believes that his daughter must submit to his commands, and this sudden refusal to follow the rules and maintain the status quo is, to him, an act of high treason. Juliet has no agency in this situation, with her father threatening to disown her and cast her into the streets if she does not obey his wishes. The fact that he is willing to sentence his daughter to a life of poverty--and a very likely early death--rather than accommodate her thoughts and feelings on the matter of her own impending wedding is telling of just how skewed the patriarchal values are at this time.