Did you not say anything during the session to the parents when the kids were acting up?
That could have covered your bases WHILE the problem was occuring.
While the kids are in the middle of the yard throwing fits, i would have gone over to the parents and had them look at the LCD on your camera while you say. "Hi Ms. So and so, im having a bit of difficulty getting some of the shots that we discussed in a level of quality that I would feel comfortable with since the kids seem to be a bit unhappy at the moment. Would you be able to assist in helping me settle them so I can capture the shots that I know we can get."
Then, when the parents try to settle them down, they will see that you ARENT going to get good shots and then you and them can come to an agreement on how to go forth (whether its a reshoot for additional money or they are happy with some of the previews on the LCD...)
Either way, you should really address the issue WHILE its happening with the client. Its not very professional to come back a couple days later after processing the photos and say. "Woops, it doenst look like I captured any shots that i feel are acceptable, we need to do it again and i might need more money"
People understand the situation more when they are currently experiencing it (i.e. seeing the kids acting up)
I just think you may have hurt the situation more by NOT communicating it clearer during the shoot. Clear, concise, and HONEST communication is extremely imortant in ALL businesses.
If YOU were the person paying for the photos, you would want the photographer to be honest and communicate clearly with you about the situation. ESPECIALLY if you decide to ask for a bit more money. As long as everyone is on the same page and understand the situation, clients have an easier time spending a bit more money and have a better experience