In a large pot of boiling salted water, cook pasta, stirring occasionally, until al dente, usually 2 to 3 minutes less than package instructions. Reserve 2 cups pasta water and drain pasta.

Meanwhile, in a large, high-sided skillet over medium-high heat, heat 1 tablespoon oil. Cook sausage, breaking up with a wooden spoon, until cooked through and browned, about 8 minutes. Using a slotted spoon, transfer sausage to a plate.

Add bell peppers, onion, ¼ teaspoon salt, and remaining 2 tablespoons oil. Cook over medium-high heat, stirring occasionally, until light golden and tender, about 8 minutes. Add garlic and red pepper and cook, stirring, until fragrant, about 2 minutes more.

Pour 1 ½ cups reserved pasta water into skillet, then add cream cheese, heavy cream, Parmesan, and ¼ teaspoon salt. Bring to a boil, stirring, until cheese is melted and sauce is smooth. Add pasta and return sausage to skillet. Toss to coat, adding more pasta water a few tablespoons at a time if needed. Add spinach and stir until just wilted, about 2 minutes.

Divide pasta among bowls. Top with more Parmesan.