In a medium bowl, whisk together balsamic vinegar, olive oil, brown sugar, garlic, and dried herbs, and season generously with salt and pepper. Reserve ¼ cup.
Add chicken to the bowl and toss to combine. Let marinate at least 20 minutes and up to overnight.
Preheat grill to medium high. Add chicken and grill, basting with reserved marinade, until cooked through, 6 minutes per side. Garnish with parsley before serving.