As you're from the US I'd concur on getting the notarized letter stating you are traveling with permission and knowledge by your parents to avoid any possible immigration hiccups. It's also good for them to put in the letter that they agree to be responsible for costs incurred by you at hotels based on standard practices of the hotel. My ex had to do this when I was 17 and traveling to see her in the UK as I would be checking into the hotel under her company's account so the hotel wanted some sort of letter saying someone would be responsible.
Another suggestion, which is a good idea to help you get solid credit established is to either get a credit card with your parent(s) as a cosigner or have your parents add you as an authorized user of one of their cards ...or both. This will give you the ability to not use up all your funds in your debit account and most hotels and such will more likely accept it. It's also a good idea in life to help establish yourself in the credit world. it seems your family trusts you enough.