1. Season flour with salt and pepper in a large resealable plastic bag. Add half of chicken; shake to coat; transfer to a plate. Repeat.
2. Warm half of oil in a large nonstick skillet over medium-high heat. Add half of chicken; cook for 4 minutes, turning once, then transfer to a plate. Repeat with remaining oil and chicken.
3. Add wine to skillet and stir with a wooden spoon, scraping up any browned bits. Cook until syrupy, 1 to 2 minutes. Add tomato sauce and reduce heat to medium-low. Add chicken; coat with sauce. Sprinkle mozzarella and parsley on top, cover skillet and cook until cheese has melted, about 5 minutes.
4. Transfer chicken to plates, sprinkle with Parmesan and serve.