Preheat the oven to 400°F (200°C). Lightly spray a large sheet pan with cooking spray and line with parchment if not nonstick.
Whisk together the balsamic vinegar and zesty Italian dressing in a bowl until combined.
Trim any fat from the chicken tenderloins, or cut chicken breasts into pieces about 1/4 to 1/2 inch thick to resemble tenders.
Place one-third of the balsamic mixture (about 1/3 cup) into a resealable plastic bag or shallow dish, add the chicken, coat well, and refrigerate to marinate at least 30 minutes or up to 6 hours.
Meanwhile, chop the broccoli into small florets and halve the baby carrots; leave cherry tomatoes whole.
Toss the broccoli, carrots, and cherry tomatoes with Italian seasoning, olive oil, garlic powder, and salt and pepper to taste on the prepared sheet pan. (If you prefer less-roasted tomatoes, add them after the other veggies have cooked 5–10 minutes.)
Roast the vegetables in the preheated oven for 10–15 minutes until they begin to soften and brown.
Remove the pan, flip the vegetables, move them to the sides of the pan, and place the marinated chicken in the center. Discard the used marinade from the chicken bag, then brush about 1/3 cup of the reserved balsamic mixture over the chicken.
Return the pan to the oven and bake another 7–15 minutes, depending on chicken thickness, until the chicken is cooked through and juices run clear. Monitor to avoid overcooking.
Serve the chicken and roasted vegetables with the remaining balsamic mixture and top with chopped fresh parsley if desired.