Fact: Corporal punishment in Florida is LEGAL. See Chapter 39 of Florida State Statutes if you doubt this is so. I know, because I used to work for DCF as a CHILD PROTECTIVE INVESTIGATOR. Hitting your child with a belt is absolutely NOT child abuse, it is a form of corporal punishment. If you choose not to use it, that is your business. But that mother's choice to use it is only an assault if you think it is, not according to Florida Law.
Having said that... if you hit the child so hard or so repeatedly that the child obtains injuries, such as bruising or swelling, then you have an abusive situation, not a corrective one.
It never ceases to amaze me how people think that their method of disciplining their child should work for everyone else. You raise your children your way and let other people raise theirs!
Personally, I would never spank my child with a belt, certainly not in public. But I do believe in corporal punishment and that is my right as a parent. And I too would have told that man to mind his own business, because how I choose to discipline my child is my business, not yours!
And just because the chain wasn't bothering you is not the issue. If the mother told the child to stop and he did not, then OH MY GOD, here's a mother that actually tried to make her child behave!!! The horror! How dare that mother just ignore the fact that her child is defying her!! I might have spanked the child with my hand, and trust me, when I was done, I would have spanked the other child for laughing at her brother! Then they both would have gone back to the hotel/home for misbehaving at the park. I would think much less of this mother if she had kept telling this child repeatedly to stop doing something and yet never made him stop it.
You can't win.... how many posts have been about people not making their children behave? Now we have a mother who was trying to make her child behave and she gets ridiculed because the OP doesn't approve of her disciplinary techniques? Six of one, half dozen of the other...
Kudos to all parents who make their children behave.. in whatever fashion that works best for them!