1. Brown Drumsticks and Veggies: Preheat oven to 400°. Pat drumsticks dry. Cook drumsticks and next 3 ingredients in hot oil in a large roasting pan over medium-high heat. Cook drumsticks 3 minutes on each side; cook vegetables, at the same time, stirring often.
2. Reserve Flavorful Pan Drippings: Bake drumsticks and vegetables in pan at 400° for 30 minutes or until a meat thermometer inserted into thickest portion of drumsticks registers 160°. Remove from oven. Remove and discard vegetables and parsley using a slotted spoon. Reserve drumsticks for another use.
3. Whisk in Chicken Broth and Stir Until Smooth: Whisk flour into hot drippings in pan, and cook over medium heat, whisking constantly, 1 minute. Gradually whisk in chicken broth until smooth. Whisk in pepper.
4. Cook Gravy to Thicken and Develop Flavor: Bring to a boil over medium-high heat, whisking occasionally. Reduce heat to medium, and gently boil, whisking occasionally, 45 minutes or until thick enough to coat the back of a spoon. Season with salt to taste.