What is the influence of the parents in Romeo and Juliet?
The parents are there to reinforce the prevailing value system, the system that ultimately leads to the young lovers' tragic deaths. A long, bloody feud has been raging between the Montagues and the Capulets longer than anyone can remember, and it's the older generation who've been keeping it going. The patriarchs and matriarchs of both families hold the respective honor of their clans in such high esteem that all else is deemed unimportant by comparison.
Unfortunately for Romeo and Juliet this would include their happiness. Love is not an important factor when it comes to getting married; marriages are strategic political alliances, not love matches. It is against this attitude that Romeo and Juliet rebel. Their heavenly love transcends the petty social customs to which they're forced to adhere by their parents. Whereas their parents remain resolutely earthbound in their insistence on social mores and traditions, including upholding the feud, Romeo and Juliet ascend to the stars in pursuit of their divine love.
For the most part, the parents' role are one of antagonists. The Montagues might be more in tune with their son's behaviors, but they do little to lessen the rivalry with the Capulets that create the climate that forces the two young people to have to hide their love away from others. The intense antagonism offered in the role of the parents have to rest with the Capulets. Both Lady and Lord Capulet are shown to represent the force of authority that does not heed the intent of the subject, in this case Juliet. The stance that both Lord and Lady Capulet take in demanding that Juliet marry Paris and the hard position taken on Juliet's own future is noteworthy. Even after Juliet has indicated that she does not want to marry Paris and Lord Capulet has essentially disowned her, he continues with plans for her marriage. In the end, the parents' roles are not that of nurterer or caretaker, as much as antagonist or obstacle that stands in the way of both childrens' happiness.