Whisk the parsley, lemon zest, lemon juice, salt, and pepper together in a large bowl. Slowly whisk in the oil until the dressing is emulsified and slightly thickened.

Add the shredded carrots and sliced radishes to the bowl and toss thoroughly to coat them in the vinaigrette.

Lay the lavash pieces flat on a clean work surface and spread the hummus evenly over each piece, leaving a 1-inch border at the edges.

Distribute the dressed carrot and radish mixture evenly over the hummus, then layer the tomato slices, avocado, and alfalfa sprouts on top.

Roll the lavash tightly around the filling, tucking the sides in as you go to secure the vegetables. Serve immediately, while the vegetables are still crisp.