I totally understand the policy. When a field trip is planned, a certain number of chaperones are needed. The students are divided up and each chaperone is given x number of students to be responsible for. Chaperone is given the rules, the times to meet, etc. Students are told that the chaperone is in charge and to listen to him/her. Teacher and chaperones, in this day and age, probably have each other's cell phone numbers as well so they can be in touch as needed.
So, then you have a mom or dad show up who isn't a chaperone and that changes the dynamics of the group the mom/dad's student is in. A younger child will want to be with mom and may not listen to the chaperone any longer. Student could get whiny, want mom/dad to take him/her to another place, buy them something, take him/her to restroom now instead of at a break time, etc. Or mom/dad could decide they want to spend more time at the bear exhibit instead of moving on to the lions as planned. Then what does the chaperone do? He/she is responsible for the child on the field trip, so either the whole group waits or chaperone has to override mom/dad which could cause conflict.
There is a reason this policy was developed/changed. If you don't like it, don't let your child go on the field trip and keep them home that day. You can take them to the zoo another time.
And, yes, the school cannot keep you from going to a public place, but you can't be a part of the field trip and you shouldn't try to crash it. Think of how difficult you could make if for your child if you are there and he/she wants to be with you but can't as he/she must stay with the group/chaperone. Do you want your child to be upset or have a meltdown or create problems? I have seen it where parents would show up and "join the group, you won't notice me" and then try and take over the group or tell Suzie that she can run over to that exhibit when she was told no by the chaperone already. It can confuse the students as to who is in charge of their group or cause the group to waste time looking and waiting for a student.
There will be plenty of field trips in the future years and you will get to be a chaperone on one of them. While you may have really wanted to go on this one, you weren't selected. So accept it gracefully.