If you do decide on the York route, there is Flamingoland just about a half hour from your destination (a few miles short of Pickering and the moors) which might prove suitable, though not especially cheap.

If choosing the Thirsk route then the Herriot Centre would be a possibility for any James Herriot fans, though that might be a little advanced for your toddler. Thirsk itself is a pleasant small town. I don't know it well enough to suggest a park, but I can't imagine it not having one.

Earlier in your trip there is an Ikea visible from the M62 near Leeds & Bradford, and easily accessible. It has the usual facilities both for snacking and play, but I have no children myself, so am uncertain whether that would be good.

Avoiding traffic jams at York on a summer weekend can be done, but you need to travel at a time few would inflict on a toddler.