1. Dry the drumsticks with paper towels, and sprinkle them with pepper and salt. Make sure to rub the spice into the meat and let it sit for a few minutes.
2. Set the skillet on medium-high heat, and add the oil and two tablespoons of butter. Heat the pan until the butter foams, and add the drumsticks. Brown each piece on all sides and move them to a plate.
3. Turn the heat down to medium-low and return the chicken to the pan. Place the lid and cook for about 20 to 25 minutes, making sure to turn each drumstick occasionally. Then, pour the rest of the butter along with the juice, zest, and garlic.
4. Toss the pan gently, so the mixture coats the drumsticks. Then, set the pan down and leave the meat to rest. Serve with the parsley sprinkled on top.

You can also watch this video for another way to cook chicken drumsticks on the stove.