Place chicken drumsticks and thighs in a shallow baking dish. Carefully slide fingers under skin to loosen it from the meat. Rub chicken all over with oil. Season with All-Purpose Spice Rub both under and over skin, patting in gently. Proceed to step 2 right away, or cover chicken and let rest for up to 24 hours. Cover and refrigerate if resting longer than 30 minutes.
Heat grill to medium-low. Oil grates; grill chicken, covered, turning frequently to prevent charring, until an instant-read thermometer inserted in thickest part (avoiding bone) registers 175 degrees, 20 to 30 minutes.