Even though we don't have a dog, I enjoyed reading the article. Most of our interactions with dogs have been pleasant, but a handful have been uncomfortable.
Usually, these uncomfortable situations happened when a parent brought both their kids and their dog to the park at the same time. This can leave the kids mostly unsupervised at the playground. Or leave a dog tied to a fence or table barking at the kids, eager to join in.
Of course, kids and dogs both need places to run around, but it can be very scary for a young child (and that child's parent) when a dog, even a friendly one, unexpectedly runs up to them. I, personally, used to pick up our son when a stranger's dog ran up to us. However, with two kids, it's not as easy to avoid an unexpected dog.
Yes, we all need to share the park. But I think many uncomfortable situations between dogs and kids could be avoided if everyone (dog owner, parent, kid, etc) had a chance to say "no" before someone else (dog, kid, stranger, etc) entered their personal space.