Remove asparagus tips and set aside. With a vegetable peeler, shave the asparagus lengthwise into thin strips. When asparagus gets too small to continue shaving, set aside.
Chop the remaining asparagus into 1/2 inch (1 cm) pieces and place in bowl with the shaved asparagus and reserved tips.
In a small bowl, whisk together the pesto, olive oil, lemon juice and zest. Pour the vinaigrette over the asparagus; scatter the asparagus tips and Parmesan curls overtop. Season with salt and pepper. Pack in a resealable container and keep chilled until ready to serve.