If you're really that concerned you should go to the hospital and have her take the morning-after pill. I believe it needs to be taken within 12 hours of the sex. But you're situation seems kind of weird. Although you should be alright considering that she just had her period. The sperm can stay alive for a few days in her, but that would still put her a couple weeks again from when she begins to ovulate. Are you sure only the tip of the condom broke up? That seems really bizarre. Usually they just rip wide open and its left just rapped around your meat but not on the top. That's been my experience in the past.

Bottom line is that you should just tell your parents you are having sex. It seems like a bad idea to you I'm sure, but odds are that even if they're pissed, they'll be glad you told them because they can help you get the protection she really needs, i.e. birth control pills.