There is a direct correlation in the play between the relationships the parents have with their children and the relationships the parents have with the outside world. More specifically, both the Capulets and Montagues have strained relationships with their children, which actually isn't uncharacteristic of the times. However, more importantly, the Capulets and the Montagues also have strained relationships with each other. Hence, one idea for a thesis is to say that Shakespeare uses the strained relationships of parents and children to show that a conflict will affect not only society, but family life as well.
In the very first scene, Shakespeare gives us a good idea of just how much the feud between the Montagues and Capulets is affecting not only the two families, but Verona's entire society. A fight that Montague and Capulet servants start quickly grows into a whole-city riot. We even learn from Prince Escalus's speech that this is actually the third riot the feud has caused. What's worse, peace had been temporarily established in the city, but now the present generation of Montagues and Capulets have started the quarrel all over again, leading Verona's citizens to take sides once again on the issue, take up their own swords again, and join in on the fray, as we see Prince Escalus explain in his lines:
Three civil brawls, bred of an airy word
Have thrice disturb'd the quiet of our streets
And made Verona's ancient citizens
Cast by their grave beseeming ornaments
To wield old partisans, in hands as old,
Canker'd with peace, to part your canker'd hate. (I.i.85-91)
In these lines, "beseeming ornaments" can be translated as "appropriate ornaments," meaning swords. What he is saying here is that the elderly citizens had figuratively buried their swords and have now had to dig them back up again to separate the two families' hatred.
Then we see that not only are the two families causing a whole-city feud, we also see that there is a great deal of friction between the children and their parents. Romeo is so distanced from his own parents, that he refuses to tell them what is troubling him, even though it's very evident he is deeply grieved over something. Juliet at first does not appear to have as troubled of a relationship as Romeo, except for having a very pushy mother. However, Juliet's decision to marry in secret leads to a great deal of tension, especially when her father decides it's in her best interest to be distracted from what he perceives to be her grief over Tybalt by marrying Count Paris. When Juliet refuses, Capulet threatens to disown her.
Hence we see that the feud between the Montagues and Capulets directly parallels feuds taking place in their own homes between parents and children. This parallel shows us just how deep a conflict can go. Even a social conflict can extend to a personal conflict, showing us that conflict, especially conflict governed by rash, uncontrolled emotions, is to be avoided.