Season flour with salt and pepper in a large resealable plastic bag. Add half of chicken; shake to coat; transfer to a plate. Repeat.
Heat half of oil in a large nonstick skillet over mediumhigh heat. Add half of chicken and cook for 4 minutes, turning once; transfer to a plate. Repeat with remaining oil and chicken.
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 mediumlow. Add chicken; coat with sauce. Sprinkle mozzarella and parsley on top, cover skillet and cook until cheese is melted, 5 minutes.
Transfer chicken to plates, sprinkle with Parmesan and serve.