Place the beets in a saucepan with water, bring to a boil over medium-high heat, and cook until tender, 10 to 15 minutes. Drain the beets, peel, and halve or quarter.
Place the sugar in a skillet and add a little water so that it looks like wet sand. Bring to a boil, and then simmer to evaporate the water out. When the sugar begins to color, remove it from the heat so that the sugar will not spatter and gently whisk in the vinegar. Return to the heat and thicken to coat the back of a spoon. Remove from the stove and stir in the horseradish. Whisk in the extra-virgin olive oil and season with salt and pepper. Toss the beets with the sauce and the parsley and serve.