Combine the celery slices, diced red onion, and chopped parsley in a mixing bowl. In a measuring cup or small bowl, whisk together the olive oil, red wine vinegar, and salt. Pour this mixture over the salad ingredients, and toss until well-coated. Cover the bowl and transfer to the fridge. Let it sit for at least an hour to let the flavors meld.
In the meantime, prepare the parmesan and walnuts. I like to use a swivel peeler on a block of Parmigiano-Reggiano to make wide shavings.
Add the cheese and walnuts to the salad after it has finished marinating. Stir well, and enjoy!
This recipe yields about four heaping cups of salad and can serve more than four if it's part of a larger spread.
If you are making this in advance, you can add the walnuts and parmesan at the same time as the other ingredients. They hold up well, so it's up to you.
The salad can marinate for up to 24 hours for maximum flavor. If you can, stir the salad every few hours while it sits in the fridge. Remove the salad from the fridge and let it sit at room temp for about 30 minutes before serving so that it isn't too cold.
To make vegan, omit the cheese. If you want this dish to be fully vegetarian, sub a vegetarian parmesan cheese for the Parmigiano-Reggiano.