Doctor Phlox kept a pet female Pyrithian bat among his many animals aboard Enterprise. (ENT: "Fight or Flight") Once, while musing about Humans' unique affinity for indulging an emotional bond with pets, he admits to himself that he has talked to his Pyrithian bat. (ENT: "Dear Doctor")
In 2152, while Captain Archer spent a night in Sickbay, the Doctor's bat got out. The two spent part of the night trying to capture it, and Dr. Phlox made a paper Pyrithian moon hawk and tried to impersonate the call of it to scare the bat. It was less then successful, but Hoshi Sato was able to catch it relatively simply. (ENT: "A Night in Sickbay")
In the Delphic Expanse, Dr. Phlox offered Malcolm Reed some of his bat's moth larvae to eat, after Reed had transformed back from being Loque'eque, and still found his stomach was sensitive to Human food. (ENT: "Extinction") He later used his bat as a distraction while he dispatched a Triannon guard in November of 2153. (ENT: "Chosen Realm")
The use of the term "Pyrithian" was originally introduced in Star Trek: Voyager in association with the Pyrithian Gorge, located on the Talaxian homeworld, in the Delta Quadrant. Since the Pyrithian bat and Pyrithian moon hawk, 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."
In the book Star Trek: Star Charts, page 44, Pyrithia was a planet in the 26 Draconis system. This system, located in the Alpha Quadrant, was a trinary star system, and was named as a stop on the Denobulan trade routes in the 22nd century.