Peel asparagus spears from top to bottom with a vegetable peeler, starting below the tips. Cut off woody ends with a knife.
Place asparagus spears in a large, wide saucepan and pour in enough lightly salted water to just cover. Bring water to a boil; add butter, sugar, and salt. Reduce heat; simmer until spears are tender and easily pierced with a knife, 15 to 25 minutes. Drain.
Divide smoked salmon between 2 serving plates. Place asparagus spears on top.
Warm oil and parsley in a small saucepan over low heat until fragrant, about 5 minutes.
Drizzle oil over asparagus. Grate Parmesan cheese on top using a cheese grater or shave into strips using a vegetable peeler. Season with pepper.