Heat the oven to 425 degrees F. On a baking sheet, toss the Brussels sprouts with 2 tablespoons olive oil and season with salt and pepper. Roast until the sprouts are nearly tender, 15 to 20 minutes.
In a medium bowl, toss the farro and turkey with the mustard and 1 tablespoon olive oil. Season with salt and pepper. Sprinkle the grains and meat over the Brussels sprouts and roast, shaking the pan occasionally, until the farro sizzles and the Brussels sprouts are golden brown and tender, 5 to 10 minutes more. Let cool slightly.
In the same medium bowl, stir together all the dressing ingredients with ½ cup olive oil. Add the arugula and parsley to the baking sheet, drizzle with dressing (you may not need all of it), and toss lightly to coat. Serve immediately.