Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally, until tender yet firm to the bite, 7 to 9 minutes.

Meanwhile, combine balsamic vinegar, 1/4 cup oil, garlic, Dijon mustard, and pepper in a small bowl. Set sauce aside.

Heat 2 tablespoons oil in a large pan over medium-high heat. Add mushrooms, squash, and onion. Cook for 3 minutes; stir in balsamic sauce mixture. Continue cooking until vegetables are tender, about 3 minutes more, adding parsley during the last minute of cooking. Drain cooked pasta, add to the pan, and toss to coat with sauce. Serve hot.