In a medium mixing bowl, whisk together the balsamic vinegar, honey, dijon, garlic, Italian seasoning, and black pepper until smooth and well combined.
Add chicken breasts to a 1-gallon resealable plastic bag. Pour balsamic mixture into the bag, over chicken. Seal the bag, removing as much air as possible.
Place bag on a rimmed baking sheet and place in the fridge 6 hours minimum or overnight.
Preheat oven to 400 degrees. Line a rimmed baking sheet with aluminum foil.
Add the broccoli, red onion, and carrots to the baking sheet and drizzle with olive oil then sprinkle with salt and black pepper. Use your hands to toss the vegetables in the oil and coat evenly.
Arrange the vegetables in a single layer around the edges of the prepared baking sheet. Remove the chicken breast from the marinade (Reserve marinade! Don't throw it out) and nestle the chicken breasts in vegetables in a single layer, and drizzle a little olive oil on top of each breast.
Place the baking sheet in the preheated oven and cook until the chicken is cooked through, and a digital thermometer registers 165 degrees when inserted into the thickest part of the chicken breast, about 18-20 minutes.
Meanwhile, while the chicken is cooking, pour marinade into a small saucepan and bring to a full boil. Adjust heat to maintain a boil (lighter boil) and cook, stirring occasionally, until the mixture becomes thick and coats the back of a spoon, about 20 minutes.
When chicken is cooked, remove from oven and drizzle with glaze. Garnish with fresh parsley and serve.