How to Make It
Preheat a grill to medium-high (400°F to 450°F). Trim carrots and turnips, and rub dry with a kitchen towel. Remove and discard tops.
Toss together carrots, turnips, olive oil, salt, and pepper in a large bowl. Place carrots and turnips directly on grates of a hot grill. Grill, uncovered, turning often, until just tender, charring slightly, 7 to 8 minutes for carrots and 11 to 13 minutes for turnips. Remove from grill, and let cool completely.
Meanwhile, prepare the Aioli: Char scallions on hot grill, 5 to 6 minutes; set aside to cool. Transfer scallions to a cutting board, and thinly slice crosswise.
Process sliced scallions, garlic, salt, lemon zest, and juice in a food processor until smooth. While machine is running, add egg yolks, processing until combined. Slowly drizzle olive oils into mixture, processing until it thickens and emulsifies. Taste for seasoning, and adjust as needed.
Slice carrots and turnips into bite-size pieces. Sprinkle with parsley, and serve with Aioli.