I agree Walton Hall is great, we really like it there.
The Garden House in Marple - volunteer run community farm with a good play area - there’s also a cafe but it’s not brilliant, but you can take a picnic. Nice to park in Marple and do the river walk to the place itself. The river is ok for paddling too - we used to do a forest school toddler session there although I think they’ve moved sites now.
Bramhall Park is also lovely, has a play area, some woodland, a duck pond - but paths are good for pushchairs or scooters/ balance bikes.
Southport - I really like it as a day out, the beach maybe isn’t the best but it’s got a nice boating lake, arcades, great play area. Shame the pier is closed at the moment, I’m not sure how long for though so worth a google.
Buxton - pavilion gardens have a good play area, a train at the weekend/ in the summer, sometimes extra things like fairground rides or bouncy castles, boating lake, cafe in the pavilion itself or a little kiosk for ice creams/ drinks on the other side, or you can take a picnic or lots of cafes in the town centre.
Some of the garden centres have little trains - High Legh near Knutsford is a good one, also has an animatronic dinosaur in the cafe plus you can look at the fish etc. Also Brookside Garden centre in Poynton - you could visit there and also drop in at Torkington park which I think has free parking, and a lovely cafe/ ice cream place backing onto it.
Lyme Park has a great play area. Still free to park if NT member.