What are the ages of your child/ren?
The big advantages of bringing your own are saving money and having the stroller outside the parks. Park strollers stay at the parks. Carrying a tired child from the bus to your resort room at the end of the day isn't fun, esp. if you are tired, too. With your own stroller, you just get off the bus/boat and pop your child into the stroller and go. With the monorails and some of the boats, you don't even have to fold your stroller - just roll it on with your child in it.
Airlines don't count strollers or car seats as baggage, and they aren't charging for these (so far!). Just gate check your stroller. Take the stroller to the gate, they will ask you to fold it and tag it, and then they take it from there. When your plane lands, they take the gate checked strollers out right there and you have the stroller when you get off the plane.
Have a great trip!