Whisk the parsley, lemon zest, lemon juice, salt, and pepper together in a medium bowl. Whisking constantly, slowly drizzle in the oil until the dressing is well emulsified. Add the carrots and radishes and toss until thoroughly coated in the vinaigrette.

Lay the lavash pieces flat on the counter and spread a thick layer of hummus over each piece, leaving a small border at the edges. Divide the tomatoes, avocados, alfalfa sprouts, and the dressed carrot mixture evenly among the lavash. Fold the sides of the lavash inward over the filling, then fold the bottom edge up and roll tightly to secure. Slice in half and serve while the vegetables are crisp.