Heat oil in a medium skillet over medium-high heat. Add onions, garlic, and 1 teaspoon salt. Cook, stirring occasionally, until onions are translucent, about 3 minutes. Stir in 1/4 cup water. Cover, and cook until soft, about 2 minutes. Let stand until cool.
Place onion mixture and cucumbers in a food processor, and process until smooth. Add yogurt, 1 teaspoon salt, and remaining 1/2 cup water, and process until smooth. Transfer to a bowl, cover, and refrigerate until chilled, at least 2 hours, or overnight.
Meanwhile, make the beets: Preheat oven to 400 degrees. Line a piece of foil with parchment. Place beets in center, drizzle with oil, and season with salt and pepper. Fold parchment and foil over beets, crimping along edges to create a packet. Place on a baking sheet, and roast until beets can be pierced easily with the tip of a knife, about 45 minutes. Remove from oven, and let stand until cool enough to handle, about 30 minutes. Rub skins off beets with a paper towel. Cut beets into 1/2-inch cubes, 1/2-inch-thick wedges, or a combination. (Beets will keep, covered and refrigerated, for up to 2 days.)
Stir parsley and lemon juice into chilled soup. Season with salt and pepper. Divide among 8 bowls. Garnish with beets and sprinkle with parsley just before serving.
The soup can be made a day in advance, but don't add the beets until just before serving or they will bleed into the soup.