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 cooked and 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 diced onion and cook for 3-4 minutes until softened.
Add minced garlic to the skillet and cook for 1 minute until fragrant.
Add diced carrots to the skillet and cook for 5 minutes until slightly tender.
In a large mixing bowl, combine the cooked quinoa, sautéed vegetables, shredded chicken, thawed peas, Greek yogurt, milk, dried thyme, paprika, salt, and black pepper. Stir until well combined.
Fold in 3/4 cup of the cheddar cheese and 3/4 cup of the mozzarella cheese into the mixture.
Transfer the mixture into a greased 9×13 inch casserole dish and spread evenly.
Sprinkle the remaining cheddar cheese, mozzarella cheese, and Parmesan cheese evenly over the top.
Bake uncovered for 20-25 minutes, or until the casserole is bubbly and the cheese on top is melted and golden brown.
Remove from the oven and let the casserole rest for 5 minutes before serving.
Garnish with chopped fresh parsley and serve warm.