In a large rectangular or oblong baking dish or container, whisk together the olive oil, balsamic vinegar, dijon mustard, garlic, ¼ teaspoon of salt, and ¼ teaspoon black pepper until well combined and emulsified.
Add the zucchini and squash to the marinade and toss to coat completely. Cover and transfer to the refrigerator to marinate for 15-30 minutes.
Preheat your grill to medium. Remove the vegetables from the marinade (keep the marinade for basting) and sprinkle with a little more salt and pepper.
Grill for about 5 minutes on the first side, flip and grill another 5-7 minutes, basting once with reserved marinade, until crisp tender with nice grill marks.
Garnish with parsley and grated Parmesan
