I guess this thread is kind of old, but it was on the first page, so whatever.
I have a spirit of competition. All I wanted to add to what has already been said was, it wasn't from just one bg. I had to run bgs a lot to get it, and I didn't get one on all my toons that ran bgs at the time. It was indeed from running battlegroup bgs. 10 to 30 minute wait, usually. Longer at night. I can't remember if I the orc warlock I got it on was 19 or a 29. He had better than average but by no means BIS gear, but that wasn't important.
I'd put the Hyacinth Macaw as a frustratingly rare pet. Guess you can buy em for a ton of gold, now and then, but farming them directly takes a tolerance for monotony I simply can't rise to, and I've done a lot of tedious grinding with this game.