I don't think the other woman handled the situation correctly with her dane. I'm guessing it was a puppy? It should've been under supervision and cut off before it even got that far. Its an owners' responsibility to be aware of what our dogs are doing in these situations and to "interupt & redirect" them if they even look like they're going to something 'wrong'.
However, that said, I also don't think that inside dog parks are places for children. Almost every dog park that I've ever been to has had rules not allowing children under a certain age at all, and over that age not without being accompanied by an adult.
I'm a mom of two(6 & 8yrs) and a dog owner, so I can see it from both points of view. Yes it would be nice to have the kids running around and playing with a multitude of dogs and not worry about anything. Sadly that is just too disneyesque for me. Children squeal and run and all too often think that they are immortal. Even the most "well trained" child doesn't listen to his/her parents 100% of the time. Dogs like to chase and pounce running squealing things. I mean, all it takes is the first time with the wrong dog.
Having lived on a unfenced farm with a "pack" of dogs for almost a decade, I learned things about dog behavior that I don't think I would have if I hadn't lived there. All it takes is for one of the dogs' prey drive to kick in and normally all the other dogs follow the first one's que before they even know who, what or why they are chasing something. Yes, I consider my dogs well trained, but they would still take off running a soon as one of them did. Luckily they all have a good recall and could be called off the chase of the cat, rabbit, deer, coyote.. etc. Though I wouldn't bet money on any dog having a 100% instant recall 100% of the time.
Then there's the fact that we have no idea how well trained others' dogs are that are in the dog park. I've heard of some dog parks being for member's only and those members having to pay a fee and pass something like a CGC to be able to join. I personally think that might be a great idea, but it would also leave a lot of dogs out. As of now, 2 of my 5 would probably be included in those left out.
Then on the dog owners side, I agree with Stella. You have no idea how other's children are going to act. Some think its perfectly fine to run screaming either towards your dog or away from it. Others were never taught how to properly behave around a dog and think its fine to pull their ears/tail/fur and sit on them or grab its paws to teach it to "give paw".
My kids are disappointed that I don't bring them to dog parks, but for me its my one on one time with whomever I'm bringing. If its an onleash thing, then sure I bring my kids, no problem.
But lots of strange dogs running around offleash, no thanks. I understand a lot of people don't agree and have never had any problems, but I just wanted to tell the reasoning behind my decision. Best of luck to you.
~moni