Preheat the oven to 350 degrees (175 degrees C). Lightly butter a 9x13-inch baking dish.

Bring a large pot of lightly salted water to a boil. Add ziti pasta and cook until al dente, about 8 minutes. Drain.

Heat a large skillet over medium heat; cook and stir onion and ground turkey until onion is translucent and turkey is no longer pink, about 7 minutes. Add spaghetti sauce and carrot; simmer until heated through, about 15 minutes. Layer ingredients into the prepared baking dish as follows: 1/2 of the ziti, Provolone cheese, sour cream, half of the sauce mixture, remaining ziti, mozzarella cheese, and remaining sauce mixture. Top with grated Parmesan cheese.

Bake in preheated oven until cheeses are melted, about 30 minutes.