The Trustees of Reservations www.thetrustees.org care about "the natural world" and "learning and discovery." Close to 136 parks are under their protection and guardianship. Bird Park in Walpole is one of them, and it has a playground. Not too many have playgrounds but they all have beautiful grounds to play on.
Bird Park http://www.thetrustees.org/places-to-visit/greater-boston/bird-park.html was my grandmother's playground in the 1920's and 30's, and now her great-grandchildren enjoy the brand new playground built this past year. Fabulous. Thankfully, it has lots of shade. Bring a picnic basket. With sand as it's cover, it features sand diggers and spinning flower cups. It's designed for children 2-5 years old. As a bonus, there is a large basketball court where little kids can ride their trikes. Lots of nice walking trails and a little pond. Oh, and recycle trash bins. Love that.