In a small bowl, whisk together balsamic vinegar, olive oil, honey, minced garlic, thyme, salt, and pepper.
Place chicken breasts in a resealable bag or baking dish, and pour the marinade over them. Seal or cover and let marinate in the refrigerator for at least 30 minutes (up to 4 hours for best results).
Preheat the oven to 425°F (220°C).
On a large baking sheet, toss the broccoli, bell peppers, and onion with olive oil, salt, and pepper. Spread them out in an even layer.
Roast the vegetables in the preheated oven for 20-25 minutes or until tender and slightly charred, stirring halfway through.
Meanwhile, heat a grill pan or skillet over medium-high heat. Remove chicken from marinade and cook for 5-7 minutes per side, or until cooked through and internal temperature reaches 165°F (75°C).
Serve the balsamic glazed chicken over the roasted vegetables, and garnish with fresh parsley.