In a large heavy-bottomed skillet or Dutch oven, heat the olive oil over high heat. Add the turkey meat in an even layer and season with the salt, paprika and red pepper. Cook until a golden crust has formed on the bottom, a few minutes. Break up the ground turkey into bite-sized pieces with your spatula and continue to cook until the meat bits are golden brown, about 5 minutes.
Add the carrots and zucchini; sauté with the meat until soft, another 5 minutes.
Carefully pour in the tomato sauce and scrape up any brown bits from the bottom of the pan. Bring to a simmer over high heat, then reduce to medium-low and allow to gently bubble until the sauce has thickened and the flavors developed, about 10 minutes.
Meanwhile, bring a large pot of water to boil. Add the gluten-free pasta and cook according to package directions. Reserve 1/3 cup of pasta water, then drain the noodles and set aside.
Off the heat, stir the crème fraiche and half the parmesan (omit for a dairy-free version) into the low FODMAP pasta sauce. Taste for seasoning and add more salt as necessary.
Toss the pasta together with the sauce, adding some of the reserved water if necessary to loosen it. Garnish with the remaining parmesan and serve immediately.