I have never seen any strollers parked outside or in any of the signature restaurants. There is certainly no area by bluezoo (have not been to Shulas) I wouldn't want to leave one outside on the boardwalk either; it's pretty congested right outside of FF with the street performers right there and I would think it might disappear.
We didnt do any signature meals so I can not help you there but we took our than 2 year old to disney world last april. We parked the stroller right outside the restaurant entryway. At a couple we were able to wheel him right to the table in his stroller because he was asleep. (We did this at 1900 park faire and Concourse steakhouse.) Being its an umbrella stroller I would probably just fold it on up and wheel it behind you to your table and stow it beneath the table or out of the way somewhere close by.