What kind of bird and what age? That will have an impact on how well they can travel.
I don't have any experience with receiving parrots by mail, but I have bought fertilized poultry eggs, day old chickens, and adult finches.
For the finches, I've purchased them from breeders out-of-state a couple times. Each time, they were shipped to me in a special shipping box that looked like this:
Inside the box is a wooden dowel perch and a water container with a sponge in it to hopefully keep the water from spilling (it doesn't work very well). Bird seed was liberally scattered all over the bottom of the box.
The first time, it took three days for the birds to reach me. The journey was stressful, but they all arrived safely and made it through quarantine with a clean bill of health. The second time, I was able to track down the package at a local distribution center and picked them up a day earlier.
I live in a rural area, so the delivery would reach the regional post office in the closest city to my address early in the morning on the day after it was shipped and then it would be held there until morning for delivery directly to my local post office. From speaking to postal workers, they said this was common practice for live animal shipments, to minimize time spent driving around in a hot vehicle.
Once the package reached the local post office, they would call me to notify it was ready for pickup and I would go get the birds. However, I didn't like leaving them in the shipping container any longer than absolutely necessary, so I made some calls and was able to pick them up early by driving to the distribution center instead.
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This isn't directly related to mailing, but do be careful when buying birds online. I have successfully purchased birds from three different people, but I have also had one place take my money and run off with it. There are a lot of scam websites out there, especially with parrots and hard to find birds. If something looks too good to be true, maybe it is. Be careful and buy (or adopt) locally, whenever possible.