How about a daddy-daughter croquet-game? My toddlers love helping their dad give that colored ball a whack around the yard. There are even alot of versions of the game aimed at toddlers... instead of wire wickets, they're shaped like animals and such. I just saw that Target has a set of toddler animal croquet in their new summer stock.
There's also hopscotch (there's rubber/foam pad versions) for use on grass.
Or how about a simple scavenger hunt? You could have parents partner with their little ones, give them a sheet of paper with simple pictures of things in nature to collect. They can look for an acorn, a leaf, a dandelion, a pretty rock, etc. Or you could bring some small trinket/cheap toys and plant them around the park (sort of like an Easter Egg hunt) and give people a list of things to look for etc. The first group to find everything wins the game.