I'm not positive but if the free ticket is specifically an adult chaperone ticket they might put an age limit on there to make sure the owner of that ticket is actually a chaperone and not a student. I went on a trip to Disneyland when I was in 10th grade and in order for someone to be a chaperone they had to be at least 25 years old, but that might have just been my high school.
Reminds me of the good old days when they had child, adult, and junior ticket prices. I remember as a kid that I couldn't wait to be a junior -- so cool! The funny thing is that I think I only went maybe once at the junior price. By the time I finally was old enough to get in as a junior they got rid of that level and switched it to just child and adult!
one way around this would be for each chaperone that gets a free ticket put in the amount that they would have had to pay in a pool and use this to buy additional youth tickets and maybe a small amount of spending money for the kids that wouldn't have been able to go due to cost.
another way to do it would be to use the money to partially subsidize some or all of the youth tickets to make it in reach of the families.
The YES programs require a specific number of adult chaperones per student ratio so that they can be responsible for the students if there were some sort of emergency, if anybody doesn't want to go on a ride, etc. So no, other students would not be able to use those tickets.