Position the rack in the center of the oven and preheat to 425 F.
In a large rimmed baking sheet, toss the brussels sprouts, parsnips, and carrots, with 3 tablespoons oil. Arrange the vegetables to form a single layer.
Sprinkle with the thyme, salt, and pepper.
Add the 1/4 cup of water and bake stirring occasionally until tender, about 45 minutes.
Transfer to a serving platter and sprinkle some fresh thyme springs on the vegetables. Serve warm or at room temperature.
Notes
Make Ahead: The brussels sprouts can be trimmed and halved and the carrots and parsnips can be peeled and cut one day in advance. Store in an airtight container or a bag with zip closure and refrigerate until ready to use.