In a large skillet over medium high-heat, warm 2 tablespoons olive oil. Season leg quarters with 1/2 teaspoon salt and 1/2 teaspoon black pepper. Add leg quarters to the pan, skin-side down. Brown chicken, turning once, 8–10 minutes per side. Remove chicken to a plate and drain off all but 2 tablespoons oil.
Add eggplant to hot pan and cook, stirring, 5 minutes. Add remaining tablespoon of olive oil, along with zucchini, onion, and garlic. Cook 5 minutes, stirring occasionally.
Raise heat to high and add white wine or water to pan, stirring to scrape up any browned bits. Add remaining ingredients except parsley and place chicken legs in mixture. Bring to boil, reduce heat to medium-low, cover, and simmer for approx. 35–40 minutes or until chicken is cooked through or until an instant read thermometer reads 170 degrees.
To serve, place some of the eggplant ragout onto plates, top with a leg quarter, and sprinkle with parsley.