Scrub the beets clean and place in a steamer basket over boiling water. Cover with a loose-fitting lid and cook until easily pierced with a knife, about 30 minutes. Allow beets to cool slightly, then slip off skins under cool running water. Remove root and stem ends and cut into 1-inch wedges.
Place the carrots in the steamer basket over boiling water. Cover with a loose-fitting lid and cook until crisp tender, about 3 minutes—I prefer mine lightly cooked.
Meanwhile, combine the olive oil, lemon juice, ginger, black pepper, and salt in a medium bowl. Set aside.
Add the cooked beets and carrots into the dressing bowl along with the cucumber and parsley. Gently toss, then refrigerate before serving.