I never cooled down any of my Parson's. Living in Southern California the goal usually was to make sure they never got too hot.
The few times I did have animal mate and lay eggs, none were put through a cooling period. Honestly I'm a believer in nature adapts.
For example, when conditions are right a given species will see its population explode. When those conditions don't exist breeding will not happen.
As humans, we can sometimes control those factors in captivity. Is slowing down the animal's biology needed? I haven't seen that. If anything I think they adapted to deal with the natural weather cycles from their home. Even their egg incubation is closely tied to those weather cycles.
A Parson in a stable temperature controlled environment will not know what time it is nor need a hibernation like period. They will just continue to need what they always do, food, light, water and all the other things we keepers give them. In the case of a female, they will produce eggs with or without a male. Keeping them cooler slows this cycle down obviously but if you make sure they have the extra calcium they need and laying sites then like an iguana, they'll just lay their eggs and go back to being a Parson in a tree afterwards.
Anyway, my .02 from the experiences I had back then. Currently with Atlas his indoor cage stays the same 79-88F year around with a night temp of about 72F