Your stroller will be your "home base" during your day in the park. It will help mama shuttle the kids between rides (the distances can be longer than they look on a map) and will be your resting spot for your things while riding on them.
Even kids who haven't ridden in a stroller in years will tire out from the pace of a Disney trip. Before heading to Florida, someone suggested renting a double stroller at the park rather than bringing my own single umbrella stroller. While my 6-year-old didn't end up spending much time in it, we appreciated the extra space. The biggest surprise for me was that we could leave our diaper bag (minus any valuables) in the stroller while we rode the rides. It was perfectly safe, and everyone in the park was doing it!