First of all, do you have kids? If you don't, that's going to make this a lot harder. Kids that age can tell a parent from a non-parent. 
Kids at 1-2 are going to be a lot different from 3-4. I think 1-2 is probably still a bit young for a playground shoot. Playgrounds can be a dangerous place for a child that age unless the parent is holding them, at which point you have more issues to deal with. You could probably do a lot better in a ball pit or one of those climbing tunnels like at Chuck-e-cheese. You can have lots of fun with that and an off-camera speedlight using E-TTL.
3-4 year olds are starting to get a bit self-conscious and many go through a fear period or can be downright hard to work with. To avoid trouble here, you can't get straight to shooting. You have to take a moment to make some sort of relationship with them. What seems to work for me is that I introduce myself and tell them that their parents asked me to take some good pics of them and that you really need their help. I use my LCD screen to show them some pics along the way. They are amazed every time the see themselves. These kids will usually pose for you and they really like it when you do something to hurt yourself for some reason. Just get on their level, be creative, have some fun, and ask for their opinion from time to time.
For the lighting in parks, I try to do the usual - shoot early morning or just before sunset, have the parent or assistant with a reflector for some fill, and go natual lighting all the way.
And of course the rules posted before about timing the shoot and meals is important. Make sure the parents have spare clothes and pack a "kid kit" similar to a wedding kit - ie needle, thread, scissors, buttons, safety pins, baby wipes, hand gel, tissues, tide-to-go stick, first aid kit (let parents do the first aid), Off, sunscreen, and of course....bubbles.
I am serious....and don't call me Shirley.
Canon 7D and a bunch of other stuff