Place a grill pan over medium-high heat or preheat a gas or charcoal grill.
Grill the corn until tender, turning every 5 to 6 minutes, about 15 minutes. When cool enough to handle, remove the kernels.
In a medium skillet, heat the butter over medium-high heat. Add the corn and season with salt and pepper. Cook until softened, about 3 minutes. Stir in the flour and cook for 1 minute. Reduce the heat to medium and add the cream. Simmer until the mixture thickens, 2 to 3 minutes. Place half of the mixture in a food processor. Blend until smooth. Pour the pureed mixture back into the skillet and add the milk, cheese, and parsley. Cook over low heat until warmed through. Season with salt and pepper, to taste, and serve.