Combine garlic and lemon juice in large bowl and set aside for at least 10 minutes. Using vegetable peeler, shave each squash lengthwise into ribbons. Peel off 3 ribbons from 1 side, then turn squash 90 degrees and peel off 3 more ribbons. Continue to turn and peel ribbons until you reach seeds. Discard core.
Whisk 2 tablespoons oil, 1/4 teaspoon salt, 1/8 teaspoon pepper, and lemon zest into garlic mixture.
Heat remaining 1 teaspoon oil in 12-inch nonstick skillet over medium-high heat until just smoking. Add squash and cook, tossing occasionally with tongs, until squash has softened and is translucent, 3 to 4 minutes.
Transfer squash to bowl with dressing, add 1 tablespoon parsley, and toss to coat. Season with salt and pepper to taste. Transfer to serving platter and sprinkle with remaining 1 1/2 teaspoons parsley. Serve immediately.