Any ideas about trying to convince my parents to let me go?

You should first mention to your parents that your boyfriends Uncle and Grandmother invited you on a trip they are taking. Tell them that you'd be really interested in going but you already know that there is very little chance of this happening. During this conversation its important to accept everything your parents are telling you, don't complain and approach the situation as maturely as possible. The result of this conversation is most likely going to be negative. At this point your best chance would be to have the grandmother or uncle contact your parents about the trip. If your parents feel that these two adults are able to keep you out of trouble and you're approaching the situation in a mature manner then they may let you go. I wouldn't mention the strengthening your relationship part as your parents probably wouldn't want that anyway! Good luck.