Yes, of course, a priest may own a pet. In fact, Pope Pius XII had a pet bird. And it’s not just diocesan priests who can have pets. If the superior allows it, religious communities can have pets. A famous story in the life of St. Therese of Lisieux is that her mother superior was overly fond of the cat owned by the Carmel at Lisieux.
Just how common it is for priests and other religious to actually own pets, I have no idea.
DISCLAIMER: Catholic Answers has turned over the archive to Catholic-Questions.org and no longer owns, manages, or moderates the forums. For additional apologetics resources please visit www.catholic.com.