I am not sure if some of this is meant to be humorous. I see the "humor" tag on here, so I am guessing so. This is an entertaining story, which is probably the point. Thanks for a fun read!
I like how you have illustrated the differences in dog parent types. As others have pointed out, there are many people who bring their dog's to the park that shouldn't be there at all because they are too aggressive. I am with the camp of folks who would rather see helicopter parents than those not involved enough. It is great your dog is well trained and has a playful nature that allows her to get along with others so well. Sadly, the bad behavior of other's have caused many of us to be mistrustful of less-involved parents such as yourself. We can't know that your dog isn't going to hurt our dog, not until we've gotten to know both of you.
To be fair, I felt the same in my human children's social situations. There are too many parents who don't supervise their children enough. It was really annoying to have to stop someone else's child from beating my child with a toy or stealing food right out of his hands because the mom felt like drinking coffee and chatting was more important than teaching her child appropriate behavior. Maybe it is balance between the two extremes that we all need.