In a mixing bowl, combine balsamic vinegar, olive oil, honey, minced garlic, dried oregano, salt, and black pepper. Whisk until well mixed to create the marinade.
Place the chicken breasts in a large resealable bag or shallow dish. Pour the marinade over the chicken, ensuring all pieces are well coated. Seal the bag or cover the dish and refrigerate for at least 30 minutes (preferably 2 hours).
Preheat your oven to 375°F (190°C) or heat a skillet over medium-high heat.
If using a skillet, remove the chicken from the marinade (discard marinade) and place it in the skillet. Cook for about 5-6 minutes on each side until the chicken is golden brown and cooked through (internal temperature should reach 165°F or 75°C).
If using the oven, place the marinated chicken in a baking dish and bake for 20 minutes, or until the internal temperature reaches 165°F (75°C).
Remove the chicken from the heat and let it rest for a few minutes before slicing. Garnish with fresh parsley if desired.
Serve warm with your choice of side dishes, such as roasted vegetables, rice, or a fresh salad.