Trim off any excess skin and fat from the chicken legs and thighs.
Place chicken in a large glass dish.
Blend the olive oil, lime juice, 2 tablespoons of the cilantro (or parsley), sazon, salt, chili pepper, garlic, and onion in a food processor or blender. Add a tablespoon or two of water if the marinade is too thick.
Pour mixture over chicken. Use tongs to turn the chicken and coat them well. Cover dish with plastic wrap, and place into the refrigerator for 3 hours.
Prepare grill for 350 degrees F.
Place chicken on grill, close lid, and cook for 8-10 minutes. Turn chicken, and cook for another 10-15 minutes, or until the internal temperature reaches 175 degrees F. Check several pieces to confirm doneness. Please note that the drumsticks might cook faster. Remove them from the grill, and keep covered until the other pieces are done.
Once cooked, remove chicken from the grill and place onto a clean serving platter. Garnish with the other 2 tablespoons of chopped cilantro or flat-leaf parsley. I like to serve mine with extra lime wedges.
Store leftovers in an airtight container in the fridge for up to 4 days. Rewarm in the oven or on stovetop for better flavor and consistency. Leftover chicken can be added to soups, stews, and casseroles.