Preheat the oven to 400 degrees F (200 degrees C). Line a baking sheet with aluminum foil.

Place chicken in a gallon-sized resealable plastic bag. Drizzle vegetable oil over the chicken. Seal the bag and massage the oil into the chicken.

Combine paprika, salt, pepper, onion powder, garlic powder, oregano, basil, thyme, and cayenne pepper in a small bowl. Mix until evenly combined. Pour spice mixture into the bag with the chicken. Seal the bag and toss to coat.

Transfer drumsticks to the prepared baking sheet, discarding the plastic bag.

Bake until chicken is no longer pink at the bone and the juices run clear, about 25 minutes. An instant-read thermometer inserted near the bone should read 165 degrees F (74 degrees C).

For extra crispy skin, set an oven rack about 6 inches from the heat source and preheat the oven's broiler. Broil chicken for 5 minutes.