Both are doable, depending on what you like to look at as you stroll.
For Stratford I would suggest parking at the Leisure Centre, then walk to Riverside, where the theatre is. Nice area to stroll around, and a gentle amble around town will take you past the Shakespeare Birthplace and the old grammar school building, if you like that sort of thing.
In Warwick, if you park at St Nicholas Park or Myton Fields, you can take in the famous view of the castle from the bridge over the Avon, then walk into town. Plenty of interesting old buildings to see in the town centre (Eastgate, Oken’s House, Lord Leycester Hospital and Westgate, St Mary’s church). Or you can park for 2 hours free at Sainsbury’s (Saltisford, by the racecourse, good for the dog), but it’s a very steep, if shorter, walk to the town centre.
Both towns have plenty of pubs, cafes etc where you can sit outside.