I would bring one with you - at 2 you will need it outside of the parks as well.
We fly from the uk and bringing the stroller is really painless, you can take the stroller right to the gate, where it is taken off you. It is returned to you at the gate at the other side, so you can use it to wheel your dd through the airport.
We took a basic Maclaren stroller and it was no bother at all on the trams, we just folded it up and put it between 2 people. (wheels on the floor, handle upright between us). I would suggest getting somethign that reclines a little as you will probably find she still naps in it, as the heat seems to make my dd more tired than at home.
Bev