Crossbows aren't considered "firearms" by the city's ordinances - it's just that the same ordinance that prohibits discharging a firearm also prohibits discharging an arrow, or a projectile from a sling:
672.14 of the Parma, OH municipal code:
(a) Except as provided in subsections (c) and (d) hereof, no person shall discharge any BB gun, air gun, rifle, shotgun, revolver, pistol or other firearm, or make use of any sling or arrow, within the corporate limits of the Municipality.
(b) No person shall, intentionally and without malice, point or aim a firearm at or toward another or discharge a firearm so pointed or aimed.
(c) This section does not extend to cases in which firearms, slings or arrows are used in self-defense, in the discharge of official duty or in justifiable homicide.
(d) This section does not extend to cases in which BB guns and other air guns, or slings or arrows, are used in the confines of dwellings, provided such use is under adult supervision and is approved by the Chief of Police.
(e) Whoever violates any of the provisions of this section is guilty of a misdemeanor of the fourth degree and shall be subject to the penalty provided in Section 698.02.