In a large bowl, toss chicken, salt, pepper, and 2 teaspoons oil. Working in batches if necessary, in an air-fryer basket, arrange drumsticks, spacing about ½" apart. Cook at 400° for 10 minutes.
Meanwhile, in a small saucepan over medium-high heat, cook garlic and remaining 1 teaspoon oil, stirring, until golden and fragrant, about 1 minute. Add ketchup, vinegar, brown sugar, paprika, and cumin. Bring to a boil. Reduce heat to medium and simmer until sauce coats back of a spoon, about 10 minutes (you should have about 1 cup sauce). Transfer half of sauce to a large heatproof bowl; reserve remaining sauce for serving.
Toss chicken with sauce in large bowl. Return chicken to air fryer, discarding any excess sauce left in bowl, and continue to cook at 400° until chicken is cooked through and charred in spots, 7 to 10 minutes more.
Arrange chicken on a platter. Top with parsley. Serve with reserved sauce alongside.
