What To Do In This Situation?

There have been a few times where I've seen a child fall over and get hurt. I feel like I want to help out and make sure the child is ok, but they're usually with their parents, so I figured they can handle the situation. Do you think it'd be a decent thing to do to still check if everything's ok even though the parent has tended to them?

If you think you could provide some assistance that the parent might not be able to, then offering that help to the parent (not to the child) would be appreciated, I'm sure. For example, "I've got some water here to clean that wound if you like." or "Do you need a plaster? I think I've got one in my bag." In all likelihood the parent will politely decline (as they'll feed that accepting help would be an admission that they're unable to provide assistance to their own child), but offering would be fine.

I'd never directly approach the child in that situation. I think it would freak-out most parents.