By the time my kids were 14, yes, I would have let them go. But we had been transitioning management of their food allergies from us to them for several years at that point. They both regularly went to Six Flags with groups of friends and no parents/chaperones starting about 7th grade.
My "rules" for them were:
1. Carry cell phone (in a ziploc if there's a possibility of getting wet).
2. Carry epi pens.
3. Only eat approved food (or food from home) and NEVER put anything in your mouth without washing your hands first. Same as eating anywhere, really.
4. Make sure at least one friend in the group knows where the epis are.
5. In addition to the regular emergency plan, let a park worker know ASAP if a reaction starts (a friend should know to do this, too).
WRT kids in the group eating peanuts, etc. People eat peanuts, etc. in public places all the time, so I always assume surfaces are contaminated to some extent. If your DS is contact or airborne sensitive, I would probably ask if the small group he's hanging out with could find alternates to nutty foods for snacking - or maybe even bring something tasty and safe to offer (assuming he will hang with a smaller group within the large group of kids going).