When his Pyrithian bat escaped from its cage in 2152, Doctor Phlox made a blue paper Pyrithian moon hawk with movable wings and imitated the call of the hawk (he mastered the call, years ago) to scare the bat. Unfortunately, he wasn't very successful and discarded of the paper hawk. (ENT: "A Night in Sickbay")
The use of the term "Pyrithian" in this series, and reference was unclear. The word was originally introduced in Star Trek: Voyager in associated with the Pyrithian Gorge, located on Talaxian homeworld, in the Delta Quadrant. Since the Pyrithian moon hawk and Pyrithian bat, two species known to the Denobulans, were presumably native to a planet in the Alpha or Beta Quadrant, it would seem apparent that this term was not related, and was simply used by two distinct cultures, such as the term "Darmok".