Preheat your oven to 400 degrees Fahrenheit (200 degrees Celsius). Line a large, rimmed baking sheet with parchment paper for easier cleanup.
In a large bowl, toss the quartered potatoes with 1 tablespoon of olive oil, salt, pepper, and paprika. Spread the potatoes in a single layer on one half of the prepared baking sheet.
In the same bowl, toss the chicken drumsticks with the remaining 2 tablespoons of olive oil, garlic powder, and half of the minced garlic. Place the drumsticks on the other half of the baking sheet, ensuring they do not overlap the potatoes too much.
Bake for 25 minutes.
While the chicken and potatoes bake, mix the remaining minced garlic with the grated Parmesan cheese in a small bowl.
After 25 minutes, remove the sheet pan from the oven. Sprinkle the Parmesan-garlic mixture evenly over the chicken drumsticks.
Return the pan to the oven and bake for another 10 to 15 minutes, or until the chicken reaches an internal temperature of 165 degrees Fahrenheit (74 degrees Celsius) and the potatoes are tender.
Garnish with fresh parsley before serving.
Notes
For crispier potatoes, you can toss them with the oil and seasonings 10 minutes before adding the chicken to the pan.
If your drumsticks are very large, you may need to increase the total cooking time by 5 to 7 minutes.
Use parchment paper or aluminum foil on the sheet pan to prevent the Parmesan cheese from sticking to the metal.