After a brief break from Minecraft. v1.8 - v1.12. I decided that a guy with an undead pirate skin needed a parrot, so off I went to the jungle. (smp6 NW outpost). After 6 hours of demolition and back filling I had created a fairly large flat area for passive mob spawning, and finally something other than an Ocelot spawned. 6 hours to get a chicken to spawn, the chicken was later killed by an ocelot. around hour 7 I saw a chicken and a sheep spawn, I moved towards them to "lower the passive mob count" I saw a parrot had spawned with them. I magic sticked the parrot killed the sheep and chicken, turned around and saw 3 more parrots spawn. I have since logged out slept ate, logged back in. now here I am 13 hours in the jungle, 4 parrot eggs and 2 ocelots heads (that don't stack). Are parrots really this rare?
First try using /entc to see how many more entities are around you, if you're close to a cap. then obviously less parrots will spawn. However, their are mob spawn conditions to be kept in mind as well, although the parrot spawn rate seems to be somewhat low in comparison. Testing this still sits on my todo list. When you say largely flat area do you mean that you also removed all the trees which block the light or did you leave them? Because light can also be an important factor in passive mob spawning (needs a light level of 9 or up, and usually it's darker in jungles). Also: was there any grass around? Because most passive mobs will only spawn on grass. Also: how far away were you? According to that same wiki page mobs will spawn at a minimal distance of 24 blocks away from you, so if your spawning area was too small then you'd reduce spawn rates. Hope this can help.
/entc comes back with an average of 50-60 hostile mobs in the surrounding jungle and subterranean caves (200 or so under cap), passive mob count hovers around 15. If you look at the map for smp6 wastelands and look in the area of the NW outpost you can see the area I've created clearly, it's more than large enough. As a matter of fact if anyone wants to capture an Ocelot come visit me, and try to catch one before I try to collect their head. I've killed 14 of them while typing this response.
In my experience a fresh jungle never loaded before will always have a higher number of parrots. I've played the waiting game too, in a jungle near my outpost.. They're just late to the game spawners... It'll always be a waiting game.
The parrot farm is now a more productive ocelot head farm that parrot, 4 parrot eggs, 5 ocelot heads.
