traffic in the city isnt too bad on the weekends, especially in the summer. i would imagine its a pretty nice drive in from the island...
i would get in early-ish, though, like ten ten thirty, eleven might be pushing it. and yeah i think you probably could find a spot on the east side then ride in around 72nd, just make sure kids have helmets, and ride safely if you go cycling.
once you get into the park, forget it, there is so much to do, especially for the little people ---rowboating, canoeing, cylcing, model sailboats, buggy rides, the cp zoo, the carousel (a must), a new amusement park with rides and games (down in the 60's on the east side), a swimming pool (north side u need swimsuits), snacks, cold drinks and iced treats on the vending carts everywhere you look, etc etc etc. and that's before you get to the bucolic setting of the park itself...full, leafy, and lush trees, well kept (and accessible) lawns and gardens, streams and little waterfalls to explore in little secret and seemingly undiscovered corners of the park...
i'm lucky, i live between cp on the east and riverside park and the hudson river (also great) on the west. but i've gotta say of all the great things the city has to offer (many way waay too costly for most), the park, free and accessible, really is every bit as great as advertised, most especially for kids, a great place to explore and discover...
btw they are closing off park ave for no cars and activiites on saturday...
http://www.nycgovparks.org/events/2010/08/07/summer-streets