Bring quinoa and 4 cups water to a boil in a medium saucepan. Season with salt, cover, reduce heat to medium-low, and simmer until quinoa is tender but not mushy, 1215 minutes. Drain; return quinoa to hot saucepan. Cover and let sit for 15 minutes. Uncover; fluff with a fork and let cool.
Cut squash into 1/8 -thick slices, some lengthwise and some crosswise. Transfer to a large bowl, season with 2 teaspoons salt, and toss to coat. Let sit until slightly wilted, about 15 minutes. Rinse under cold water and drain well. Pat dry with paper towels.
Whisk grated Parmesan, zest, juice, and vinegar in a medium bowl. Gradually whisk in oil. Season dressing with salt and pepper.
Combine squash, quinoa, parsley, walnuts, and basil in a large bowl. Pour dressing over; toss to coat. Garnish with shaved Parmesan.
Per serving: 350 calories, 29 g fat, 3 g fiber Nutritional analysis provided by Bon Appétit