Preheat the oven to 350°F (175°C).
In a medium saucepan, combine the rinsed wild rice and chicken broth. Bring to a boil, then reduce heat to low, cover, and simmer for 45-50 minutes until the rice is tender and liquid is absorbed. Remove from heat and set aside.
While the rice cooks, heat olive oil in a large skillet over medium heat. Add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and sliced mushrooms to the skillet. Cook for 6-8 minutes, stirring occasionally, until mushrooms are browned and tender. Remove from heat.
In a separate medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes until lightly golden, stirring constantly.
Gradually whisk in the milk and continue cooking, whisking frequently, until the sauce thickens, about 4-5 minutes.
Remove the sauce from heat and stir in the sour cream, dried thyme, salt, and black pepper until smooth.
In a large mixing bowl, combine the cooked wild rice, mushroom mixture, shredded chicken, and creamy sauce. Mix until well combined.
Transfer the mixture to a greased 9×13-inch baking dish. Sprinkle the shredded cheddar cheese and grated Parmesan evenly over the top.
Bake uncovered in the preheated oven for 25-30 minutes, until the casserole is bubbly and the cheese is melted and golden.
Remove from oven and let rest for 5 minutes. Garnish with chopped fresh parsley before serving.