Combine sugar and lemon segments in a small bowl; let stand at room temperature 1 hour. Remove lemon segments from bowl with a slotted spoon (reserve lemon juice mixture for another use).
Thinly slice beets crosswise into 1/8-inch-thick rounds. Arrange beet slices, lemon segments, and parsley on a large platter. Sprinkle with walnuts and goat cheese. Drizzle with oil. Sprinkle with salt and pepper.