At least, so says this biologist. Basically, the idea is that instead of being a splinter off of raptors (hawks and eagles) which specialized for speed, falcons are actually a splinter off of parrots who specialized for speed and meat eating.
The article goes on at length about how shocking this is, but having been very into studying falconry in my youth and having once had a pet parrot (well, a cockatiel, but it’s in the parrot family), I’m actually not very surprised. Eagles (mostly) look kind of like large hawks, but falcons just don’t look much like hawks at all. They actually look more like short-tailed parrots. This is especially true of their wing shape. Falcons have long, narrow wings where the flight feathers stick together forming a (mostly) solid surface. Hawk’s and eagle’s flight feathers stick out independently, looking almost like outstretched fingers. Parrot’s wings look extremely similar to falcon’s wings, with the flight feathers touching each other.
Granted, looks aren’t dispositive, which is the point of the original article. I just think it’s worth noting that it goes out of its way to emphasize the ways that falcons look like hawks but not how they look dissimilar.
And it should be noted that the results of convergent evolution are, in the end, convergence. That falcons are more closely related to parrots than to hawks means very little; you can train hawks and falcons for falconry (hunting), but you can’t train parrots to hunt. Falcons don’t talk like parrots do, and you interact with them much more like hawks than like parrots. Hawks, eagles, and falcons are all fairly solitary creatures.
So while it’s fascinating trivial that falcons are more closely related to parrots than to hawks, it’s not actually useful information. You still shouldn’t name your gyrfalcon polly, or offer it a cracker.