Using a food processor, pulse together the parsley, chile, lime juice, garlic and 3/4 teaspoon salt. Scrape down the sides. With the machine on, pour in 1/4 cup olive oil. Transfer to a small bowl, stir in the heavy cream and refrigerate until ready to serve.
Preheat the grill to medium-high. In a large bowl, toss the drumsticks with the remaining 2 tablespoons olive oil; season with salt and pepper. Arrange the chicken on the grill and cook until grill marks appear, about 3 minutes on each side. Lower the heat to medium-low, cover and cook, turning occasionally, until the chicken juices run clear, about 20 minutes. Serve with the dipping sauce.