I have had the opporunity to have participated in many group dog training classes over a long period of time. I have observed many dogs with problem behaviors overcome those behaviors through training. In some dogs, the behavior or urge to do the behavior was changed or eliminated. In others, the dog learned to ebey and ignore his desire to do something inappropriate. These behaviors included dog agression, pet agression and child agression. It is actually not difficult to overcome agression problems in dogs. You do not even have to go to expensive training, just going and comitting to a dog training program and putting in the time and effort can mend or control agression behaviors. I would say that the only variable that I saw that ever predicted which dog would still be disobedient at the end of the course was the owners. Owners who came to class and refused to learn new concepts of dog training (Like clicker or reward/incentive) or didn't bother to practice and maintain the discipline at home. I am by no means an expert in dog training, but give me any dog and I promise that I can improve that dogs problem behaviors in 6 weeks or less. There are some exceptions, dogs with behavior problems based on poor socialization in puppyhood are very difficult to train, and abused dogs can be quite unstable. However, in the case of the dog at the park, I think a lot of wild assumptions are flying around about the dog and the owner.
First, we don't know if she realized that the park did not allow dogs. Sometimes people are just oblivious.
Second, we don't know if the dog had been agressive before, so lets take the owner at her word. Now she does know.
Third, is the dog genuinely agressive? We don't know. No adult really was able to witness the scene, (We can't trust the dog owners account, and the OP didn't see it) and the child probably was terrified and can't make an accurate judgement wabout it. We'll call it agression, but what kind? Excitement, fear, agression? We don't know.
Finally, this really may have been the first time the dog acted that way. We don't know if it is or is not, but maybe the owner is going to adress the problem correctly, by taking the dog to training. Since it was a minor act and according to the owner the first sign of agression, the dog is very likely to overcome this problem if the owner takes it for training. Lets hope she does that. It would be the right thing to do.
OP, this is what I think about the incident as it relates to you and your child. If I were you, I would try to take the lady at her word about the dog and her apology, and be glad this happend. First, because your child was uninjured, but also because now the dog owner is aware of a potential problem and can now take care of it before it can happen again. If it happens again, it could be more serious.