Answer: Ciliocytophthoria, a.k.a. detached ciliary tufts from ciliated epithelial cells.
Florida Fan mentions that "Considering how many things we breath in daily, it is quite marvelous they sweep them out more efficiently than we can imagine." Very true. Our ciliated epithelial cells do an excellent job as part of the "mucociliary escalator" that removes inhaled debris from the respiratory tree. Unfortunately, various inflammatory states including asthma can cause the apical ciliated tufts of the respiratory cells to detach, and more than one investigator has mistaken these for ciliated parasites.
Of note, the only ciliated parasite of humans is Balantidium coli, a large organism (40-200 micrometers) with circumferential cilia and a classic "boring" motility. In comparison, ciliary tufts are much smaller (10-15 micrometers in diameter), have cilia arising from a flat plate-like structure, and have a beating motion in fresh preparations.