You technically dont, but you will need parental permission to travel internationally (all of you except the 16 year old I believe will). Real question though is, why would you want to travel alone- thats a huge risk. I am not sure which form that is to get signed- but I know you need one. My parents would NEVER allow me to go overseas without an adult at that age, and didnt want me to go alone even when I turned 18 but didnt have a choice to let me go.
Also- how are you planning to get around- if your staying in one city like New York, then your fine, but other then that, you wont get to see much because the train system here isnt near as good as yours if your in Europe. (It does the trick, but its expensive and also not all major cities have local trains- just busses which are pretty sketchy (especially at night). Youll have to stay near the city centers, and thats expensive to do. Our cities arent walkable for the most part like yours are if your in Europe.