The footpath that runs almost from the hospital/nestle to Osbaldwick (Route 66/Foss Island path) is all flat and only crosses a road once and loads of ways to get down to it. It’s also nice and green for most of the year and St Nicholas Fields is a nice stop halfway along with lots of green space and a playground etc. They’ve done loads of work on it recently so the surface is loads better than it was too!
Askham bog is a nice loop with the pushchair. Rowntree park is good too. I’ve been out to Fountains Abbey and had a really nice long walk with my pram there