I would say it depends on the park. At MGM, the strollers were a canvas like fabric (similar to jogging strollers) but at MK the strollers were a hard plastic, which would be pretty uncomfortable.
Just my opinion, but with a 1 year old I would definately bring your own stroller. We still bring our own and my daughter will be 4 in August. The park strollers are only available "in the parks". So, you wouldn't have one when walking to/from the busses, shopping at Downtown Disney, walking the Boardwalk, or walking around your resort, etc. When you throw in the diaper bag, cameras, stuff you purchased that day, the stroller comes in real handy.