Whisk the parsley, lemon zest, lemon juice, salt, and pepper in a large bowl. Slowly drizzle in the oil while whisking constantly until the dressing is thickened and emulsified.

Add the shredded carrots and sliced radishes to the bowl and toss until every piece is well-coated in the vinaigrette.

Lay the lavash pieces flat on a clean counter and spread the hummus evenly over each piece, leaving a small border at the edges. Arrange the tomatoes, avocados, alfalfa sprouts, and the dressed carrot mixture in even layers over the hummus.

Fold the short sides of the lavash inward over the filling, then fold the bottom edge up and roll tightly into a secure cylinder. Slice each wrap in half on a bias and serve immediately while the vegetables are at their peak crunch.