Preheat the oven to 375°F (190°C). Lightly grease a 9×13-inch casserole dish.
Heat olive oil in a large skillet over medium-high heat. Add the chicken pieces and cook for 4-5 minutes until lightly browned but not fully cooked through. Remove chicken and set aside.
In the same skillet, add chopped onion and cook for 3-4 minutes until softened. Add minced garlic and cook for 1 minute until fragrant.
Add the uncooked rice to the skillet and stir to coat with the onion and garlic mixture. Cook for 1-2 minutes.
In a medium bowl, whisk together chicken broth, milk, sour cream, lemon juice, lemon zest, dried thyme, salt, and black pepper until smooth.
Transfer the rice mixture to the prepared casserole dish. Evenly distribute the browned chicken pieces on top.
Pour the creamy lemon sauce over the rice and chicken in the casserole dish. Gently stir to combine and ensure rice is submerged.
Sprinkle shredded cheddar cheese evenly over the top of the casserole.
Cover the casserole dish tightly with aluminum foil and bake for 35 minutes.
Remove the foil and sprinkle grated Parmesan cheese over the casserole. Return to the oven uncovered and bake for an additional 10 minutes until the cheese is golden and bubbly and the rice is tender.
Remove from the oven and let the casserole rest for 5 minutes before serving.
Garnish with chopped fresh parsley and serve warm.