I wouldn't think that the rays came out of the Marine Park area, at least I would hope not.
The impression we are given is that the park is well maintained and the rules of fishing are strickly enforced.
I don't think permits would be required to catch the rays. Not to be crass, but I would imagine a simple fishing liscense would do it...your catching fish. Just depends on jurisdictions where they were taken, who knows, could have been in the Caymans, Florida, the Gulf or Bahamas for that matter. They are not endagered animals, so I don't think taking them for this purpose is a big legal issue.
If the purpose of the venture was public education and the animals had full-time medical staff and facilities, aquarists or zooligists taking care of them like in a zoo or aquarium, then maybe it would be ok. The shallow cage close to shore worries me because of wave action causing safety issues for the fish.
At the aquarium we use the animals to educate guests on the importance of taking care of the planet and thus it's inhabitants. For instance we have a sea turtle from a south Texas turtle rescue with 2 partially missing flippers. One from an apparent shark bite, the other was amputated because plastic was wrapped around and imbeded into it and the fin had to be taken off to get the plastic off the animal.
By letting people SEE what their litter can do, or physically touch an animal that they may only ever get to see on TV, you can make an emotional connection that hopefully they will carry with them and think next time they have the choice of making "green" decisions or not. It's much better than just preaching facts and gloom & doom.