Peel the asparagus, and thinly slice on a bias. Season a pot of boiling salted water with the sugar. Simmer the asparagus in the water for 3-4 minutes, until al dente.
For the dressing, rinse and dry the parsley. Remove the leaves and finely chop. Mix the vinegar, lemon juice, 2-3 tablespoons of asparagus cooking water, and the oil. Season to taste with salt and pepper. Mix the parsley into the dresssing.
Drain the asparagus and drain well. Mix the asparagus into the dressing. Season to taste, and serve the salad at room temperature.