Bring a large pot of salted water to a boil. Add the pasta and cook according to the package instructions, about 8 minutes. Drain and set aside.
Season the chicken with 1 teaspoon salt and some pepper. Dredge in the flour and shake off the excess. Heat the oil and butter in a large saucepan over medium-high heat until the butter melts. Carefully place the dredged chicken into pan. Add the grated garlic. Cook until the chicken is golden brown and cooked through, 4 to 6 minutes. Transfer to a plate and set aside.
Pour the cream into the same pan and bring to a simmer. Stir in the Parmesan. Add the pasta to the mixture and toss to combine. Add the cooked chicken and gently toss. Garnish with the parsley and serve immediately.