In a large pot of boiling, salted water, cook the pasta until al dente, 8 to 10 minutes; drain.
Meanwhile, season the chicken with salt and pepper. In a large skillet, heat the olive oil over medium heat. Add the chicken and cook, turning once, until browned, about 8 minutes; transfer to a plate.
In the same skillet, cook the garlic over medium heat until golden, about 1 minute. Add the wine, bring to a boil and cook for 1 minute. Stir in the tomatoes and simmer until thickened, 8 to 10 minutes. Return the chicken and any juices to the skillet, turning to coat. Top each piece of chicken with some of the mozzarella and 1 teaspoon parmesan, cover and cook until the cheese is melted, about 2 minutes; transfer to a platter.
Add the pasta and parsley to the skillet, toss and season with salt. Sprinkle with the remaining parmesan and serve alongside the chicken.