I can't quite believe we're having this argument. Pedestrians, including kids, have absolute priority on footpaths, regardless of whether the council has painted a white line down the middle and designated it shared use.
Kids have every right to run around a park as erratically and unpredictably as they want - there is no reason for them to be "reined in" by their parents (unless there is a pond or something). Cyclists simply shouldn't figure on the list of hazards in a park. If you're cycling in a park in such a way that you are a hazard, then you're doing it wrong.
As a cyclist, it is my responsibility to avoid pedestrians, not the other way around. If I cycle carefully and at an appropriate speed, the chance of a collision is virtually nil.