In a small bowl, stir together the ingredients for the balsamic marinade (olive oil, balsamic vinegar, garlic, brown sugar, soy sauce, Dijon, and some freshly cracked pepper).
Cut the mushrooms, bell pepper, zucchini, and onion into one-inch cubes, or as close to that size as possible. Leave the grape tomatoes whole. Place the vegetables in a gallon-sized zip top bag and pour the marinade over top. Massage the bag to mix the vegetables with the marinade. Let them marinate for 30 minutes at room temperature, flipping the bag occasionally to redistribute the marinade.
While the vegetables are marinating, begin the rice. Combine the uncooked rice, water, minced garlic, and salt in a medium pot. Place a lid on top and bring the pot up to a boil over high heat. As soon as it reaches a boil, turn the heat down to low and let the pot simmer for 15 minutes. After 15 minutes, turn the heat off and let it rest for 5 more minutes, then fluff with a fork. Allow the rice to cool slightly.
While the rice is cooking, prepare the lemon parsley mix. Pull the leaves from about 1/2 bunch of parsley. Finely chop the leaves until they appear minced. Use a zester, microplane, or small-holed cheese grater to remove the zest from one lemon. Combine the minced parsley, about 1/2 Tbsp lemon zest, and olive oil in a small bowl.
When the rice has cooled just slightly, add the lemon parsley mix and fold gently until the rice is coated in parsley, lemon, and olive oil. Avoid vigorous stirring, as this can make the rice pasty.
Adjust the oven rack to the second position from the top and preheat the broiler on high. Cover a large baking sheet with parchment. Spread the marinated vegetables out over the surface of the baking sheet. If they do not cover the sheet in a single layer, use two baking sheets to avoid over crowding.
Broil the vegetables about 6-8 inches from the flame for 10 minutes, or until they achieve a subtle char on the edges. Every broiler is slightly different, so keep a close eye on them and turn the baking sheet as needed. For more even broiling, stir the vegetables half way through.
Fill each bowl with a scoop of lemon parsley rice and a pile of the broiled balsamic vegetables.