In mortar or using the flat side of a chef’s knife, mash the garlic to a paste with a pinch of salt. Put the paste in a small bowl (or keep it in the mortar) and whisk in the lemon juice. Let sit for 5 to 10 minutes and then whisk in the olive oil.
Using a mandoline or a sharp chef’s knife, cut the squash diagonally into very thin (1/16- to 1/8-inch) ovals. Put the squash in a medium bowl, season with salt and pepper, and gently toss with about two-thirds of the vinaigrette.
Combine the arugula, parsley, chives, and capers in a separate bowl, season with salt and pepper, and toss with just enough vinaigrette to lightly coat. Taste both the squash and herbs and adjust the seasoning with salt or pepper if necessary. Layer about a third of the squash in a shallow bowl or platter, scatter about a third of the arugula mixture on top, and sprinkle with a third of the grated Parmesan. Repeat the process with the remaining squash and arugula mixture, sprinkling each layer with grated Parmesan. For garnish, use a vegetable peeler to shave long strips from the chunk of Parmesan onto the salad. Serve immediately.