How to Make It
Preheat oven to 375°F. Toss together sweet potato, onion, chiles, parsnips, cumin, 1½ tablespoons of the oil, and ¼ teaspoon of the salt on a large rimmed baking sheet. Bake until vegetables are tender and lightly charred, about 40 minutes, stirring once after 20 minutes.
Arrange tortillas on a work surface. Sprinkle ¼ cup cheese over half of each tortilla; top evenly with roasted vegetable mixture. Fold tortillas in half over filling.
Heat ½ tablespoon of the oil in a large nonstick skillet over medium-high. Add 2 assembled quesadillas to skillet; cook until cheese has melted and quesadilla is lightly browned, 1 to 2 minutes per side. Transfer to a plate. Repeat procedure with ½ tablespoon oil and remaining 2 quesadillas.
Whisk together honey, black pepper, 1 tablespoon of the lime juice, and remaining ½ tablespoon oil and ¼ teaspoon salt in a medium bowl. Add arugula and radishes; toss gently to coat. In a separate small bowl, stir together yogurt and remaining 1 tablespoon lime juice.
Cut each quesadilla into 4 wedges. Serve with salad topped with yogurt-lime mixture.