Personally, I don't, but I see it a lot, and I know it winds a lot of pedestrians up.
I came across a dad cycling with his kids the other day (one kid on a bike with stabilisers) and they were all on the pavement, he got a proper dressing down from a lady waiting at a pedestrian crossing when they all approached to cross. Although I don't agree with pavement cycling in general, for small children I don't think it's necessarily a good thing to have them cycling on the road until they're of an age to be able to do that safely. But the woman who had a go was saying that none of them should be on the pavement, even the children. I did think perhaps that dad should have cycled on the road beside the kids though.
When I was a kid we cycled around on the pavements all the time, talking when we were under 10 here, around the estate we lived on. OK, you can take your kids to the park to cycle around, but not everyone has a car and bike rack to be able to transport the kids and bikes to the park or somewhere suitable they can cycle off the road.
Do you ride on pavements/allow your kids to ride on pavements? Has it ever been a problem?