Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Reserve 1 cup cooking water, then drain.
Meanwhile, heat the olive oil in a separate large pot over medium-high heat. Add the kale and cook, stirring, until wilted and the liquid evaporates, 2 to 3 minutes; remove to a bowl. Add the heavy cream and 1/2 teaspoon salt to the pot. Bring to a simmer and cook, stirring, until thickened, 5 to 8 minutes.
Add the pasta, reserved cooking water and parmesan to the sauce and stir until creamy. Season with salt and pepper and stir in the kale. Divide among bowls and season with pepper.