Nope, you just park the stroller and pick up another where you get off (if there is one like you need). Any stroller you bring onboard the train has to be able to fold up. The rented ones don't fold.
As Deb&Bill stated. The train offers an exchange system. You would take your name tag and your personal belongins and leave your stroller wherever you board the train. When you arrive at the station you are going to they have replacement "exchange strollers" there and you would just take one of those. They are usually pretty stocked (especially at Toon Town)...but if you happen to arrive at a station that does not have a double that you could take they will get you one over on the next train (3-7 minutes) or they will direct you to where you can pick up another one
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