Preheat your oven to 400 degrees F (200 degrees C). Line a large baking sheet with parchment paper for easier cleanup.
In a large bowl, toss the potatoes with 2 tablespoons of olive oil, salt, and pepper. Spread the potatoes in a single layer on the prepared baking sheet.
Roast the potatoes for 15 minutes while you prepare the chicken.
In the same bowl, combine the remaining 2 tablespoons of olive oil, minced garlic, and dried oregano. Add the chicken drumsticks and toss until they are fully coated in the garlic mixture.
Remove the baking sheet from the oven. Push the potatoes to one side, creating space for the chicken. Arrange the coated drumsticks on the other side of the sheet pan in a single layer.
Return the sheet pan to the oven and bake for 20 minutes.
Remove the pan again. Sprinkle the grated Parmesan cheese evenly over the chicken drumsticks.
Continue baking for another 10 to 15 minutes, or until the chicken reaches an internal temperature of 165 degrees F (74 degrees C) and the Parmesan forms a light crust.
Garnish with fresh parsley before serving.
Notes
For crispier skin, place the chicken on a wire rack set inside the baking sheet before adding the Parmesan.
If your potatoes are cut larger than 1 inch, you may need to start them roasting 5 to 10 minutes before adding the chicken.
Use freshly grated Parmesan cheese for the best melting and crust formation.