Pat the chicken dry with paper towels.
Combine olive oil, garlic, lime juice, paprika, cumin, oregano, salt, and pepper.
Rub the marinade evenly over the chicken.
Cover and refrigerate for at least 2 hours or overnight.
Preheat the oven to 425°F (220°C) or prepare a grill over medium-high heat.
Roast the chicken for 35–40 minutes until golden and cooked through.
If grilling, cook for 30–35 minutes, turning occasionally until the internal temperature reaches 165°F (74°C).
Meanwhile, add cilantro, jalapeños, garlic, mayonnaise, yogurt, lime juice, olive oil, Parmesan, salt, and pepper to a blender.
Blend until smooth and creamy.
Adjust seasoning or lime juice to taste.
Serve the hot chicken with the green sauce on the side.
Garnish with extra cilantro and lime wedges if desired.