Preheat the oven to 350 degrees F (175 degrees C). Lightly grease 2 baking sheets.
Combine parsley, carrot, mozzarella cheese, and oil in a bowl. Stir together flour, bran flakes, and baking powder in a separate bowl. Add parsley mixture to flour mixture and stir well. Gradually mix in water until everything is well combined. Add a couple tablespoons more water if needed to create a moist dough. Knead dough for 1 minute.
Roll dough out to 1/3-inch thickness and cut out 36 treats using a dog bone-shaped cookie cutter. Place on prepared baking sheets.
Bake in the preheated oven until golden brown, 20 to 30 minutes. Turn off oven. Remove baking sheets, turn treats over on the baking sheets, and return to warm oven for an additional 30 minutes.
Remove baking sheets from the oven and cool treats on the baking sheets for 1 minute. Transfer to a wire rack to cool completely. Store cooled treats in an airtight container.
