Preheat the oven to 375°F (190°C).
In a medium saucepan, combine the rinsed quinoa and chicken broth. Bring to a boil over medium-high heat, then reduce heat to low, cover, and simmer for 15 minutes or until the quinoa is tender and the liquid is absorbed. Remove from heat and fluff with a fork.
While the quinoa cooks, heat olive oil in a large skillet over medium heat. Add the diced onion and cook for 3-4 minutes until softened.
Add the minced garlic, diced carrots, and diced red bell pepper to the skillet. Cook for another 5-6 minutes, stirring occasionally, until the vegetables are tender.
In a large mixing bowl, combine the cooked quinoa, sautéed vegetables, cooked chicken, dried thyme, dried oregano, salt, and black pepper. Stir well to combine.
Add the shredded cheddar cheese, shredded mozzarella cheese, and plain Greek yogurt to the mixture. Stir until evenly incorporated.
Transfer the mixture to a greased 9×13-inch casserole dish and spread evenly.
Sprinkle the grated Parmesan cheese evenly over the top of the casserole.
Bake in the preheated oven for 20-25 minutes, or until the cheese is melted and bubbly and the casserole is heated through.
Remove from the oven and let it rest for 5 minutes before serving.
Garnish with chopped fresh parsley and serve warm.