For the fries: Preheat oven to 400 degrees F. Place the squash sticks on one baking sheet, and place the parsnip sticks on another. Drizzle both with the olive oil, and sprinkle with salt and pepper. Cook the squash sticks for about 5 minutes, and cook the parsnips until golden and crisp, turning once or twice, about 30 minutes.
For the dip: Combine the basil, garlic lemon juice zest in a food processor fitted with a steel blade pulse until smooth. With the machine on, slowly pour in the olive oil. Blend until smooth. Transfer to a small bowl and add Greek yogurt; mix together. Serve immediately or refrigerate for up to 1 week.