Personally I wouldn't use parrots as a model for chickens in much of any behaviors. Eye pinning in parrots can be very specific to the individual bird. For example, my parents had multiple parrots and there was one where it was only ever a warning and the only warning you'd get before he bit extremely hard. That was just the one bird though. It can also be affectionate excitement in some parrots, or a sign of nervousness, but it's quite variable. If a parrot I didn't know eye pinned at me, I would be cautious around it because of the ambiguity. Eye pinning in chickens has seemed much more consistent to me in terms of the context that leads to it.
Others' chickens may be different, but I have only ever observed eye pinning as a form of affectionate excitement in my chickens, such as observing chickens that like each other eye pin at each other. I don't think I've ever seen it involved in truly aggressive interactions in mature birds, like pecking order disputes or rooster fights. Nearly all my chickens do it when I say their names, and all my chickens are very friendly with me. So, I would interpret eye pinning when a chicken's name is said to mean two things: the chicken knows its name and it likes you.
