Preheat oven to 400°F. Place squash halves, cut sides down, on a baking sheet lined with parchment paper. Bake until tender when pierced with a fork, about 40 minutes. Remove from oven; let cool 10 minutes.
While squash bakes and cools, heat oil in a Dutch oven over medium-high. Add mushrooms; cook, stirring occasionally, until lightly browned, 6 to 8 minutes. Add turkey; cook, stirring often, until turkey is no longer pink, about 3 minutes. Add carrots, onion, and garlic; cook, stirring often, until vegetables are slightly softened, about 5 minutes. Stir in tomatoes, salt, and pepper. Increase heat to high; bring to a boil. Reduce heat to medium-low; simmer, covered, stirring occasionally, until vegetables are tender, 25 to 30 minutes. Uncover; simmer, stirring occasionally, until thickened, 10 to 12 minutes. Remove from heat. Stir in parsley, basil, and vinegar.