Rinse the brown rice, then combine with 2 cups water or low-sodium broth in a medium saucepan; bring to a boil, reduce heat, cover, and simmer until tender, about 35–40 minutes. Fluff and set aside.
Heat the olive oil in a large skillet over medium heat. Add the chopped carrots and sauté 3–4 minutes until slightly softened.
Add the ground beef and ground turkey to the skillet, breaking up the meat with a spoon. Cook until no pink remains and liquid has mostly evaporated, about 6–8 minutes.
Stir in the peas and green beans and cook another 3–4 minutes until heated through and tender-crisp. If using frozen vegetables, cook until fully warmed.
Combine the cooked rice with the meat and vegetable mixture in the skillet. Stir well to combine and cook 1–2 minutes to meld flavors. Sprinkle in dried parsley if using.
Remove from heat and let cool to lukewarm before serving to your dog. Portion as appropriate for your pet's size and dietary needs.