In a large, shallow baking dish whisk together the soy sauce, brown sugar, honey, and tomato paste. Remove 1/4 cup of the mixture; set aside. Add the chicken drumsticks and tenders to the remaining mixture in the dish. Turn pieces to coat with sauce. Cover and chill up to 8 hours.
Adjust grill for indirect grilling. Brush grill rack with oil. Test for medium heat over unlit side. Remove chicken drumsticks from dish; add to unlit side of the grill. Cover and grill for 35 minutes or until done (180°F), turning and brushing with reserved glaze once. Remove the chicken tenders from the dish (discard remaining mixture in dish). Add to the lit side of the grill the last 8 minutes of grilling or until done (170°F), turning and brushing once with glaze. Remove chicken from grill to a platter and serve.