IMO - You did the right thing in this instance, and it sounds like this child has serious psycological problems that NEED to be addressed. I'm wondering if this is comming up in school, if it's a constant at your house, there's sure to be an issue at home as well. I'm a very outspoken person, and in a case such as this I would:
1. Talk to my own children about the dangers of bullying and how to seek out adult help in the situation, the story should be re-told from the victims mouth to the parent of the bullying child if possible.
2. Have an honest to goodness tounge thrashing with the child (as you clearly did)
3. Talk to the parent. You've spoken timidly to this person, but if she is your friend, and she considers you likewise, than formalities from the public park aren't as necessary. While parents may not enjoy having their child put down by others, it seems important in this case that the mother know how you feel. I'd express your concern about how her child is hurting your children, offer your help, suggest formal counseling, be a friend. If this was my son and I was blind to it, I'd expect someone in my circle to speak up! I know this can lead to a disaster, but if this woman can't see the issue, and her child isn't welcome in your home, than risking the friendship might be worth it to save the child from lifelong problems.