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.