Bring 4 cups water to a boil in a medium saucepan. Add quinoa. Season with salt, cover and reduce heat to medium-low. Simmer until quinoa is tender but not mushy, about 12-15 minutes. Drain the quinoa and return it the saucepan. Cover and let quinoa sit for 15 minutes. Uncover; fluff with a fork and let cool.
Cut squash into 1/8"-thick slices, some lengthwise and some crosswise. If you have a mandolin, this is the perfect time to dust it off. The thinner you can slice the squash, the better. Transfer the squash to a large bowl, season with 2 teaspoons salt, and toss to coat. Let sit until slightly wilted (a fancy way of saying the squash will soften a bit), about 15 minutes. Rinse under cold water and drain well. This step is important, otherwise your squash will be too salty. Pat dry with paper towels.
Whisk grated Parmesan, zest, juice, and vinegar in a medium bowl. Gradually whisk in oil. Season dressing to taste 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.