I got caught at my girlfriends house (we are both 16) and our parents don't want us to date anymore, so she's grounded. What's should I do to convince them to let us date? Keep in mind that I got caught a few days before I met her dad.

This is a very delicate and sensitive subject.  Personally, I would (if you can do so sincerely) issue an apology to her parents.  After all, parents are protective of their children, your significant other is a minor, and perhaps, with more rules, they may let you date again if they think you are mature and realize you were wrong.

Remember that every family has different beliefs and values.  Depending on what your significant other's family's are and what you got caught doing they may be very upset about what you did.  Try to look at it from their side and be respectful of their beliefs even if you do not share them  This will help in making you seem more mature and like a better potential significant other to the parents.

Then, depending on how well apologizing went, express your feelings for you significant other and your desire to date them to the parents.  Ask what they rules would be.  If you are willing to follow their rules, accept and date within those rules.  Their rules may seem silly but remember, you don't want to mess it up with the parents again or you guys might not be able to date anymore.

If they are unwilling to reconsider, try to respect their rules.  After all, she is 17, soon she will be 18 and if the two of you still like each other, you can date without having to ask the parents.  (Although do keep in mind what your parents would think as your significant other will be 18 and you will still be a minor.  Your parents could always charge her if the two of you engaged in sexual behaviors while you are a minor in many states).