As Kojiro said, unless the gear specifies what job it is for, be it by the job(s) capable of equipping it or by specifying "Avatar", "Wyvern", or "Automaton", that gear will affect any kind of pet. For example, the Moepapa Stone, which gives Pet: Haste +5%, is can be equipped by Beastmaster, Summoner, Dragoon, and Puppetmaster, and will give the pets of those jobs Haste +5%.
As for those gear pieces that can be equipped by jobs that don't natively have pets, remember that, outside of Dragoon's Wyvern, you have access to pets in some form if you sub those jobs. You are capable of using Charm with /Bst, and you can call out Avatars and Automatons as /Smn and /Pup, respectively. That and Square Enix tends to add jobs to gear in bulk, even if the stats for that gear may only be suited for one job. Beastmaster, for example, is oftentimes added to the same gear that Warrior is added to, so unless it's a JSE or JSE-esque piece, usually a piece of gear added that is good for Beastmaster will probably also have Warrior on it just because of SE's penchant for doing that.
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