Just back from York, but live near Edinburgh. I felt York was less walking than Edinburgh and there was loads to do (although my DSs are older now). The only think I thought that was bad about York for a little one in a buggy was the narrow streets, I felt a bit sorry for the parents trying to push buggies through the crowds in the narrow streets.
We went to the Castle Museaum, Yorvick, Dig and a trip on the river in a boat, it was all great fun but perhaps more for older children, however I thought the shops and the little streets and markets were lovely in York and I would have been perfectly happy strolling round them. I'm sure we passed a big park when we were on the boat trip and there is always feeding the ducks. However you are right about the zoo in Edinburgh that's great for little ones.
The weather in York last weekend was fantastic and currently the weather up here, very near Edinburgh is lovely, although not quite as warm as it was in York.
Don't forget princes street gardens right in the middle of Edinburgh - you could probably spend a whole day there! There is a great playpark, a cafe, a fountain, a fairground carousel, behind the bandstand there is a railway bridge with clear plastic sides where toddlers can watch the trains, there are loads of lovely green slopes for rolling down and having picnics, and it's also right next to all the shops (so when my DS has a nap in the buggy I can have a lovely browse in Boots or whatever).