Preheat oven to 425°F. Wrap beets in heavy-duty foil and seal edges well. Roast for 1 1/2 hours. When cool enough to handle, remove greens, slip off peel and halve crosswise; cut halves into 8 wedges. Toss with 1 tsp. oil and salt and pepper.
Toss parsnips with salt, pepper and 1 Tbsp. oil on a rimmed baking sheet. Roast in oven, turning once, until fork-tender, 30 minutes.
Toss Brussels sprouts with remaining oil, salt and pepper on another baking sheet. Roast, turning twice, until softened, 15 to 20 minutes.
Arrange on a platter. (Vegetables can be made several hours ahead and kept, covered, at room temperature.) Sprinkle with parsley.