Combine the balsamic vinegar, honey, brown sugar, soy sauce and garlic cloves in a large resealable bag. Add the chicken to the bag and seal. Marinate in the refrigerator for 2 hours.
Preheat the oven to 450 degrees. Line a rimmed baking sheet with aluminum foil. Top with a cooling rack. Remove the chicken from the bag, reserving the marinade, and arrange the drumsticks on the cooling rack. Bake until the skin is caramelized and dark in spots, about 30 minutes.
Meanwhile, place the leftover marinade in a small saucepan, uncovered. Bring the marinade to a boil, then reduce the heat to a simmer and cook over low heat until thick, about 15 minutes. Use a pastry brush to brush the cooked marinade on the chicken. Return to the oven for 10 more minutes.
Place the chicken on a serving platter and sprinkle with the sesame seeds and the chopped parsley.
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