1. To make the salsa verde, place the anchovies, capers, and garlic in a food processor and pulse until finely chopped. Add the carrot tops, rosemary, oregano, thyme, red chile flakes, and lemon juice and process until finely chopped. With the processor running, gradually pour in the olive oil through the feed tube and puree until very smooth. Season to taste with pepper and more lemon juice, if you like. (The sauce can be stored in the refrigerator, tightly covered, for 2 days before serving.)
2. To make the roasted vegetables, preheat the oven to 400°.
3. Place the potatoes, carrots, and parsnips in a large bowl with the olive oil and salt and pepper to taste. Toss to coat, then spread the vegetables out in a single layer on two rimmed baking sheets with the potatoes cut side down. Roast until the vegetables are tender and golden, 30 to 40 minutes, flipping the potatoes and stirring the other vegetables halfway through and removing vegetables as they are done.
4. Serve the vegetables warm, drizzled with some of the salsa verde.