Heat oven to 425°F; foil-line 18-by-13-inch rimmed sheet. Pat drumsticks dry; coat with 1 tablespoon oil, paprika, garlic powder, pepper, and 1/2 teaspoon salt.
Toss potatoes with remaining oil and salt. Arrange chicken and potatoes separately in one layer.
Roast 25 minutes; turn chicken and potatoes. Roast 10 minutes more; transfer fully tender potatoes to clean plate.
Brush chicken with barbecue sauce. Roast 8 to 12 minutes until tacky and chicken at least 165°F, preferably 175 to 185°F for tenderness.
Rest 5 minutes. If needed, warm reserved potatoes in clean open pan section during last 2 minutes of cooking. Serve 2 drumsticks with one-quarter potatoes per portion.