Preheat oven to 350 degrees F.
In a baking pan toast walnuts in 1 layer in middle of oven until golden and fragrant, about 10 minutes. Cool nuts and coarsely chop.
In a large saucepan, simmer beets in water to cover until just tender, about 30 minutes or longer (especially if using medium or large beets, which can take up to an hour to cook). Drain beets. When beets are just cool enough to handle, peel and quarter them. While beets are still warm, in a bowl toss with vinegar. Beets and nuts may be prepared 2 days ahead and chilled separately, covered.
Toss beets with nuts, oil, parsley, and pepper and season with salt.