Don't parrot perches count as an "add on"? If so, I don't believe those count towards your total available lockdowns (add-ons such as gozas didn't used to count as lockdowns from what I recall). I'd have to double check on Test Center.
However, apart from the lockdown issue, because the parrot perches block movement you would not be able to utilize all the possible tile area of the "normal" floor areas of a full 18X18 grid house, but you could get close. You would need a minimum number of tiles dedicated to supports to hold your upper floors, and if you placed a teleporter tile in one corner of each floor, and carefully worked your way back towards the teleporter leaving one clear tile in front of the teleporter to be able to move back and forth to trigger the teleporter, you could have quite a large number of parrots places, and a house you could not use for anything else, if that is what you are asking.
-Skylark