If there is a dog in the parks, it has probably been entered as a service dog. A service dog has been trained to perform some sort of task that assists a person with a disability.
There isn't anything in the law about multiple dogs although it does stretch the bounds of credibility that a single disabled person would need several service dogs. Possibly there were multiple disabled persons in the group, the dogs were in training (federal law doesn't equate service dogs in training with fully trained service dogs, but local laws might), or each dog had been trained to perform a separate task to assist the same person. Because no proof is required, I guess all they can do is admit the person with 7 service dogs once they say they are all service dogs.
Therapy, comfort or emotional support animals are not service animals, as they have not been trained to perform a specific task to assist a person with a disability. Don't know whether Disney excludes emotional support animals from their parks, but they would not be required by law to admit them.
The turkey in the photo is not a service animal. It is an emotional support animal. There is a regulation that applies specifically to airlines that says they must admit emotional support animals, but the owner may be required to provide something like a note from a doctor stating they need one (with a service animal, the business cannot require any proof that the animal is actually a service animal.)
However, in actuality you can use a website to contact a "doctor" who will diagnose you based on your answers to some online questions and provide you with a letter that says you need an emotional support animal. Cost between $100-200.
If this dog is some sort of online "celebrity" it is also possible that they allowed it in for a photo shoot, like they did with the Grumpy Cat.