Preheat the oven to 375°F (190°C). Grease a 9x13-inch baking dish or spray with cooking spray.
Place crushed Ritz crackers in a shallow dish and stir in 1/4 teaspoon salt, 1/8 teaspoon pepper, and 1 teaspoon dried parsley.
Put 1/2 cup sour cream in a separate shallow dish and the 3 cups grated cheddar cheese in another shallow dish for coating.
Working with one chicken half at a time, dip it into the sour cream to coat all sides, then press into the shredded cheddar to adhere cheese, and finally press into the cracker mixture until well coated. Repeat with all pieces.
Arrange the coated chicken pieces in the prepared baking dish in a single layer. Cover the dish with foil and bake for 50–55 minutes.
Remove the foil and bake an additional 10 minutes, or until the coating is golden and the chicken reaches 165°F (74°C) and juices run clear.
While the chicken finishes, combine the 10.5-ounce can of cream of chicken soup, 1/8 cup sour cream, 2 tablespoons butter, and 2 tablespoons milk in a small saucepan over medium heat; stir until warmed and smooth.
Serve the baked chicken with the warmed sauce spooned over the top and garnish with fresh parsley.