Congratulations on having a wise grandmother, and yes, mites could cause a bird to do that, but you would probably see the bird interested in the early stages of mutilation if mites are involved.....if you were to suspect mites, place a sheet of white typing/copy paper in the bottom of the cage, then get up before dawn/sunup/outdoor light, in the morning and, while watching your white paper, have the lights turned on...if you have mites, you will see them scurrying to the dark corners/recesses of the cage.....
In the event you do see mites, you will need to remove everything from the cage, including any detritus and treat the cage, particularly all crack, crevices & hidden openings for mites, before allowing your birds back in.....
Actually, mites may not be your problem, because most varieties of mites, are not fond of parrot blood.....
Here's some extra info you might be interested in:
Bird Mites | Plant and Insect Diagnostic Clinic
