In the bottom of a salad bowl, whisk together the oil, vinegar, lemon juice, salt, and honey with a fork. Add the red onion and season with pepper. Let stand 10 minutes or longer.
To core pears as the pros do, halve them lengthwise and scoop out the cores with a melon baller. Make a little v-cut to remove the stem from the bottom ends, and, with a knife, lift out any of the long stem. Cut crosswise into slices.
Add the arugula and toss to mix. Add the pears and walnuts and toss lightly. Taste, adding more salt, pepper, or vinegar, if desired. With a vegetable peeler, shave wide Parmesan curls over the top. Serve right away.