Season chicken on both sides with salt and pepper to taste.
Heat 1/2 tablespoon of the oil in a large skillet over medium-high heat; add chicken and sauté for 3 minutes per side or until well browned; transfer to a plate and keep warm.
Reduce skillet heat to medium and add remaining oil.
Add apples, onion, and thyme; cook and stir until apples and onion are softened, 2 to 3 minutes. Add cider and broth; bring to a simmer and cook until slightly thickened, about 3 minutes.
Return chicken and accumulated juices to the skillet.
Reduce heat to low and simmer until chicken is cooked through, about 4 minutes; transfer chicken to a warmed platter.
Stir sour cream and parsley into the skillet and season with salt and pepper; spoon sauce over chicken.