This is a really good question. There are many theories as to who is at fault for the tragic end to Romeo and Juliet, but I think looking at the parents, is a good start. Many people had a hand in what transpired to the two lovers, but the actions of their parents could have easily led to what happened.
The first way they could be at fault, is that they should have just let the relationship play out. We have to remember that Juliet was a very young girl, and this was her first experience with love. It is a good possibility that she was just infatuated with Romeo. If her parents had let the relationship play out, they may have realized there was no need for worry. If Romeo's parents had done the same thing, they may have realized that their son's attention might have been drawn elsewhere.
Secondly, if both sets of parents had let their feud end, then there would be no worries over the relationship. The grudge had gone on for many years, and if the parents had been adults and ended the hatred with themselves, then their children wouldn't have had to carry around their parents feud. The two families were wealthy and in a good social standing. There would have been no reason why Romeo and Juliet couldn't have been together. Instead of letting a petty feud run their lives, they could have forgiven each other, and then their children would have been free to see if their relationship was real.
Lastly, if Capulet hadn't been so insistent on Juliet marrying Paris, then maybe the tragedy could have been avoided. By tying to force Juliet to marry a man she didn't love, Capulet just sent Juliet right into the waiting arms of Romeo. If her father would have acknowledged her feelings, then we might have seen a happier ending. Instead, Capulet was intent on forcing his daughter to marry a man she didn't love, and forcing her away from the man she did love.
In the end, there are several people who could play a role in the tragedy. Everyone had a reason why Romeo and Juliet shouldn't be together, but none of them took into consideration how Romeo and Juliet felt. If they had taken their feelings into the mix, maybe the whole tragedy would have never had happened.