Presumably you've read Nixxie's thread?
Definitely go to Betty's for lunch/the entire afternoon. Don't be put off by the queue of empty nesters and the fancy waitress outfits, they are really child friendly, they have a kiddy menu, lovely big toilets with spacious change area and potty, they give you toys and wipes at the table for babies and the service is really efficient (and the waitresses are lovely). We always eat there when we go to York. It's heaven, no hardship for you at all!
There is a nice park around the Museum, not sure if there is a play area there though, not taken the girls to the Museum but from my pre-child memory it's a bit stuffy.
There's some nice gardens round the Minster with part of the city wall, depends how fearless you are if you go on the walls or not, there are a lot of areas with no protection from falling off. Don't walk the whole way round, the nicest bit is from the river to the Minster.
Yorvik is very popular with LOs, you can prebook on the internet to avoid the queues. We went to York Castle Museum last time we were in York but I think the girls were still a bit young (they were tired though so might have enjoyed it more if they'd had a bit more energy).