In Frontierland, under the railroad tressle, there is a small playspace that is one of our favorite resting areas for DD. It has a small slide (perfect for a 16 month old) and a playhouse to walk/climb through. The flooring is also rubber playground padding, however it will scrape if one falls and skins an arm or knee. I love it, it's totally in the shade and great place for Mom or Dad to watch the littl eone whiel the other rides Big Thunder Mountain Railroad or Splash Mountain.
Also, Fantasyland is a good area to let a toddler out of a stroller and walk. The ridea are pretty close together (unless you plan on riding Small World after Pooh which are on opposite ends), but a little one can walk through while holding a grown-ups hand. Usually we park our stroller at Philharmagic and walk around Fantasyland.
Also there are times, I will find a bench somewhee and just sit. I'll let DD enjoy a snack and let her out of her stroller to be able to play and explore around me. She's 3 now, but when she was younger (her first trip was at 15 months), I kept an eye on her and controlled her rein. We have a joke in our family that when she was 22 months, she was so excited to go beyond 10 steps and throw somethign in a trash can. Our joke was, forget the rides at Disney, it's the trash cans that take center stage.