Use your vegetable/potato peeler to shave the asparagus. Shave each piece of asparagus individually by laying it flat on your cutting board, using one hand to hold one end and using the other hand to shave it. Place asparagus in a salad bowl.
Chop shallot and parsley. Sprinkle over the shaved asparagus.
Shave the parmesan using the same vegetable peeler, and add it to the salad.
Make the dressing by combining all dressing ingredients in a small bowl and whisking them until smooth and emulsified, or by adding them to a jar and shaking.
Toss salad with dressing, add salt & pepper, and serve immediately.