Remove the stem end of each onion and carefully slice off the brown part of the root end, leaving the root intact. Peel the onion. Stand each onion root end up on a cutting board and cut the onion in wedges through the root. Place the wedges in a bowl.
For the dressing, combine the lemon juice, mustard, garlic, thyme, salt, and pepper in a large bowl. Slowly whisk in the olive oil. Pour the dressing over the onions and toss well.
With a slotted spoon, transfer the onions to a sheet pan, reserving the vinaigrette that remains in the bowl. Bake the onions for 30 to 45 minutes, until tender and browned. Toss the onions once during cooking. Remove from the oven, and drizzle with the reserved dressing. Sprinkle with parsley, season to taste and serve warm or at room temperature.