"Bit of a joke in the common genre, a poor farmer sows a heap of seed and goes to bed. He wakes up in the morning and thinks the seeds have grown into cockatoos as the field is full of them. Hence a cocky farmer is an unsophisticated small farmer."

"The only ones picking the fruit on our tall tree were the very raucous cockatoos."

I don't know the origin of cocky, but the image I get is a farmer who - despite his efforts - only manages to raise cockatoos.

Ring-necked parakeets are increasingly common in the UK.