Put the red onions, lemon zest and salt in a bowl, let sit 5 minutes. Rinse the onions and lemon zest and pat dry. Add the lemon juice, rinsed onions and lemon zest to a large bowl and toss.
Using a vegetable peeler, peel the asparagus stalks into ribbons, coarsely chop the remaining center pieces and add to bowl with the onions. Add the lettuce, parsley, and olive oil and serve immediately.