There are brokers or import agents in most countries that assist and enable smooth importation, making sure all the documentation is there, following the process, because it PAYS to have a human paying attention to the movement and care of the animal, a broker or import agent is a hugely useful critter. And they don't do it for free.
There are the fees to the breeder, as was said, fees - testing, certification. The quaratine costs. The agent or broker fees and shipping costs.
The only way family on one side helps is that it is cheaper to ship an animal accompanying a travelling person than separate. But then the travelling family member is paying for the flight- aren't they. And then the animal still goes through customs (more $) and then into quarantine, the rest of the $$$.
So no, there aren't a lot of easy shortcuts. Importation of any animal is NOT cheap. They're careful of birds because of the avian flus etc. Even if it's paranoia it's increased the income of both sending and recieving countries significantly.
So yes, figure far more than $1000 and into the $3000 range depending on country of origin, fees, certificates, shipping, quarantine, customs(both countries) and the cost of a good purebred bird worth importing.
Can you tell I've imported dogs?