Using 2 large sheets of heavy duty aluminum foil, lay 4 beets on top of each sheet. Drizzle beets with olive oil, and wrap the foil around them in a package, seal the edges well. Place them on the upper rack of a preheated barbecue or in the oven at 350°F (177°C) and roast until tender, approximately 1 hour. Let cool and peel if desired. Cut into large cubes.
In a large bowl, combine vinegar, walnut oil, salt and pepper. Taste to adjust. The sweetness of the beets will come through, so a sour vinaigrette works very well. Toss in the beets and onion. Just before serving, top with crumbled feta and chopped parsley. Serve at room temperature.