Heat oven to 350 degrees. Heat oil in a large ovenproof pot over medium-high heat. Season chicken generously with salt and pepper. Cook until golden on all sides, about 5 minutes, and set aside.
Reduce heat to medium and cook leeks until tender, about 4 minutes. Add vinegar and scrape up brown bits with a wooden spoon. Add parsnips, sage, stock, and seared chicken and its juices and bring to a boil.
Cover and transfer to oven. Braise until chicken and parsnips are very tender, about 50 minutes. Serve over rice.
When braising poultry, use darker, more fibrous bone-in cuts, such as legs and thighs, for better flavor.