I am quite new to avian surgery, but my sources suggest more than 12 hours before surgery is excessive. For example, in a baby parrot that would be of comparable age to a young chicken, 6 hours seems to be plenty.
I would agree that taking water away more than 12 hours is not a good idea, they become lethargic, especially if they are small for attempting the surgery. Also, unless it is plenty warm enough, I think they get cold when they go through the fasting period.
I must say though, if you are comparing a parrot to a chicken, I think it is like apples and oranges. They are both birds, but I think a chicken is probably bigger and hardier than a caged parrot. I agree with Jeff that it is not the age, but the size you should consider when deciding when to caponize. I will also add that I think that the books written on caponizing were written, many of them or taken their information from, 80-100 years ago. Chickens today are not the same. A lot has changed in the way they are bred, many being large hatchery birds who have gone after the golden egg count, the qualities of the heritage breeds have been lost for a large part through those methods. The birds, I believe, were probably larger and more robust that many we see today - that is, unless you are getting them from breeders who are focusing on maintaining the integrity of the breed, then you may see them in the size more in line with the standard for the breed.
I have Minorcas, which are a smaller framed, lighter weight breed than say my Cornish or my Australorps, for those I have to add a couple of weeks to the recommended 4-5 weeks before I will even think about cutting into them, and then I pick them up and feel what they are feeling like before I start looking at a day to put them on the table.
Yes, I have only done a few, but every day I go through all of my birds and I look at them and evaluate. I have Minorcas who are 2-months old, which I know some would say they are getting big to do the procedure, but I have about 20 of them so far that are going to need to go and I don't want to put them down and go to waste, and I don't want to raise them to adolescence and have them fighting in the pasture and beating up the hens in the process. So, if they need to be 2-months old to get the deed done, then so be it.
That is my soap box for the day.