In a 9×13-inch casserole dish, whisk together the olive oil, sesame oil, honey, soy sauce, minced garlic, grated ginger, and freshly ground pepper until combined.
Add the chicken drumsticks to the dish and toss to coat each piece thoroughly with the marinade.
Cover the dish with plastic wrap and refrigerate to marinate for at least 2 hours or up to overnight.
Preheat the oven to 375°F.
Remove the plastic wrap and place the dish in the preheated oven. Bake the drumsticks for about 45 minutes, turning once halfway through to promote even browning.
Bake until an instant-read thermometer inserted into the thickest part of a drumstick registers 175°F and the skin is nicely browned.
Garnish with chopped parsley and serve with the pan sauce.