For all the erudite (and occasionally amusing) explanations for the nickname, most of you Sassenachs seem to be unaware that in Scotland a budgerigar is invariably known as a budgie and so it is no surprise that the name rapidly became popular at Glasgow. As I pointed out at Post No 4 - it's a small noisy bird.
Due to the chequered history of the ATP at introduction, it became known as "the Parrot" - on the grounds that it was bigger than a Budgie and always sick.
To be fair, they settled down in service and even commanded the affection of their crews.