Preheat the oven to 375°F (190°C) and line a baking sheet with parchment paper.
Rinse the drumsticks briefly under cold water, drain, and pat dry with paper towels.
In a bowl, combine the dry rub and salt if the rub does not already contain salt; rub the mixture generously over the drumsticks to coat all sides.
Place the seasoned drumsticks on the prepared baking sheet in a single layer.
Bake for 20 minutes. Remove from the oven, brush the drumsticks with the olive oil, turn them over, and return to the oven.
Bake an additional 10 minutes, then check doneness by cutting a slit or using an instant-read thermometer; the internal temperature should reach 165°F (74°C). If not done, bake 5–10 minutes more until cooked through.
Serve the drumsticks with the BBQ sauce on the side or brush the sauce on before serving.