Well, what route will you take?
I have done a few different routes, I like heading east from Santa Cruz at Watsonville on the 156 and get on the I-5 as soon as practical.
In the first leg, both Casa de Fruta (commercial location that has kids attractions) and the San Luis Reservoir (public park) are good before you get on the I-5 South.
On the I-5,there are some public rest stops that have some grassy areas and picnic table, along with restrooms.
http://www.californiarestareas.com/interstatei5restareas.html
Commercial options for the most part are small areas just designed for travelers with a few motels, restaurants and gas stations, but some of them will be something like McDonalds with an indoor playground.
Just after the I-5 merges with the 99 south of Bakersfield, there are the new Outlets at Tejon, which has some kid friendly areas as part of the area.
After that,i would look at Castaic as the last "small" area before you get into the suburb area of Los Angeles, and then your choices are limited, but you are getting close to the DLR.
And then I hope you have a GPS with live traffic or someone who can use a smartphone app with live traffic, as there are multiple route options starting just past the Santa Clarita area.