Welcome to the Daily Halacha. Are pets muktzeh (set aside) on Shabbat? The answer is yes. Animals are muktzeh on Shabbat, whether a person has a parakeet, as they call it, or a dog or a cat. Ba'alei chayim nechshavim l'muktzeh b'Shabbat. Animals are considered muktzeh on Shabbat. You can't even pet them because when you pet them, you're really moving them, and you're not allowed to move muktzeh on Shabbat. There are exceptions to the rule if a person needs a dog for therapy, which does happen sometimes. Then the dog can be considered not muktzeh if you need to walk with it or help it a little bit.
Are you allowed to feed animals on Shabbat? That's something else. The answer is yes. If you own the pet, you own the animal, and you're responsible to feed it, you're allowed to feed pets on Shabbat. But you're not allowed to move pets on Shabbat. So from here we learn, Rabotai (my teachers), even though sometimes it might seem normal, a person should not own dogs because dogs are muktzeh on Shabbat, unless you're planning not to touch your dog on Shabbat. But if it's for therapy use, ask your local rabbi because there are exceptions to the rule.