Spread flour in a shallow dish. Season chicken pieces with the salt and pepper. Dip chicken in the flour, turning to coat evenly and gently shaking off excess.
In a 5-quart Dutch oven, heat oil over medium-high heat. Add chicken pieces; cook about 6 minutes or until browned, turning occasionally.
Remove chicken from Dutch oven; set aside. Drain off fat, reserving 2 tablespoons in the Dutch oven. Add carrots, celery, onion, and garlic. Cook about 5 minutes or just until onion is tender, stirring occasionally.
Stir in tomato paste. Add broth, wine, and vinegar; bring to boiling. Add tomatoes and thyme. Return chicken to Dutch oven. Bring to boiling; reduce heat. Simmer, covered, for 60 to 70 minutes or until chicken and vegetables are tender.
To serve, place chicken, vegetables, and cooking juices on a serving platter. Sprinkle with Parmesan cheese and parsley.
I added a lot more pepper than it called for at the end. I also used a can of drained petite diced tomatoes instead of fresh plum tomatoes since we're not big tomato chunk fans. Husband and I both enjoyed it.