Among the most ornate animal traits in nature are the angle-dependent (e.g. iridescent) structural colors of many fishes, damselflies, birds, beetles, and butterflies. Though we now have a solid understanding of the mechanisms that create angle-dependent coloration in several groups, we know little about whether pigmentary colors reflect light in an angle-dependent fashion or if similar or different mechanisms govern angle-dependent reflectance from pigmentary versus structural colors. Here for the first time we describe non-iridescent angle-dependent coloration from the tail and wing feathers of several parrot species (Aves: Psittaciformes). We employed a novel approach—by calculating chromatic and achromatic contrasts (in just noticeable differences, JNDs) of straight and angled measurements of the same feather patch—to test for perceptually relevant angle-dependent changes in coloration on dorsal and ventral feather surfaces. We found, among the 15 parrot species studied, significant angle dependence for seven of our eight feather JND parameters. We then measured micro-scale features on each side of feathers, including size and color of barbs and barbules, to attempt to predict interspecific variation in degree of angle-dependent reflectance. We found that barb height, plumage-color type (e.g. melanin, structural), and differences between barb-barbule coloration (measured using Euclidean distances) were the strongest predictors of angle-dependent coloration. Interestingly, there was no significant phylogenetic signal in any of the angle-dependence models tested. These findings deepen our views on the importance of microscopic feather features in the production of directional animal coloration, especially in tissues that are colored predominantly by pigments and appear to be statically colored.