Hold up just one second.
You were at the park, a public park or a private park?
The mother objected. Asked to delete the images, I would've probably have done the same thing.
The skipped beat here is - what you're doing is journalism. You're in public, and according to law you have a right to shoot whatever you want whenever you want - as long as it isn't from private, to private or privacy protected property/items/people.
The hint here is she stopped freaking out when you got out your phone. What you should've done is called the police and reported her for harassment, and calling someone a pedophile is criminal slander - she could've been charged and you would've walked away. It's a knob thing to do but technically she would've been ****ed in that situation.
If you take a photo in public propety it has to be a group shot involving more elements than the kids. If you take a shot of a playground, the subject is the playground. If you take a shot of a 100 % crop of a kid's face - yeah that's problematic. It's a shame the moderator took the photo down, because it has everything to do with deciding if you were in the wrong or not. If it was a kid, and the central subject was the kid, then you'd have an issue. If you are photographing a person under 16 you need a release to display commercially or in print, granted by the guardian of the person.
That said, taking a picture of them and doing nothing with the photo isn't illegal, strangely. It's when you print/reproduce it (and maybe post it online im not sure about that) you have the issue.
Does anyone think that a journalist goes up to everyone in the frame they shoot ? Or just even one person? A journalist asks for permission before submitting photos of the president and the president grants him a release? Doesn't happen.