From the beginning I knew the mother was lying. She was probably so shocked that he reacted to swat him on the behind on impulse, that she took it all out of proportion. The kid must have gotten a bloody nose from all that running around and struggling.
Dude if you're a parent and you can't control your kids, to where you have to chase them down in a public place, play tug of war, and the kid is hitting strangers, don't take them out, simple.
And I know some people here have children and might say "oh you don't understand", but I do. My mom would beat my ass in public even if she had to, so of course I'd behave.