Yeah, you can totally bring your own stroller. We are. There are things to consider though. Using the bus service you will need to CARRY the stroller onto the bus. So you will want one that collapses well and isnt too heavy! I also would want to make sure that you stroller pushes easy, as the parks are really big. But the big advantage of bringing your own to me is: not waiting in line for the stroller at the parks, being able to use your stroller at the resort and airport, and having one that your child is use to. If my son were older, we would likely rent one at the parks rather than bring our own, but since he will only be 23 months, I would be more comfortable with ours. We are taking our graco travel stroller, it reclines and has a basket and sunshade, but is really light and compact. We have used it as a second stroller since he was born so he is very comfy in it.
I tried to copy and paste a photo of the strollers in the parks for you, but then I got concerned about copywrite laws, but there are pictures of the strollers at www. allearsnet .com