Most of the parks in the canyon are federal I believe. I doubt there will be many there so I think paying at the gate is plenty likely. I'd see what you can find out about them though, it's early in the season.
Not much for state parks locally and I haven't looked at local parks. Sportsman's is too close to the homeless community for my taste. (They're the predominant wildlife around town.)
If you're at SOZO, it's on the far south side of town. (Closer to Yakama than Yakima, but I don't think the tribe has any RV parks!

)
They're joking about
staying in Eastern Downtown... but they're not joking about the mini-seattle the city fathers have created down there. It's a MESS. Avoid it entirely.
Great area, great kids' sports community, have fun!