I couldn't find any information on whether birds have a fusiform gyrus (which I had never heard of prior to this thread). But they definitely have the ability to recognize individuals based on their physical appearance. There are some birds who can recognize specific flockmates by the patterns on their feathers, in species where there's a lot of individual variation in feather color.
Some species can recognize individual humans too, and parrots obviously fit into that category. But the funniest example is the experiment where people put on masks and harassed wild crows, then came back later with the masks on to see if the crows still recognized them. The crows did.
http://www.nytimes.com/2008/08/26/science/26crow.htmlThere usually isn't a lot of ultraviolet light inside a house, so that probably isn't it. The birds just simply know what we look like and can tell when somebody is not us. It makes sense from a survival point of view. They're prey animals and need to know whether the other animals moving around in their environment are safe or not. They've spent enough time with you and your dad to know that you're safe. But they're not so sure about other people, and a smart bird will be cautious. A dumb bird will instantly be friendly to everyone they meet, which makes people love them. But these birds would last about 30 seconds in the wild.