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.
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.
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.
Test Kitchen Tip: To make ahead: Cool gravy 45 minutes. Cover and chill up to 3 days. Add a few tablespoons of broth, and reheat over medium heat.