Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally until cooked through but firm to the bite, about 12 minutes. Drain and transfer to a pasta bowl.
Combine garlic and olive oil in a cold skillet. Cook over medium heat to slowly toast garlic, about 10 minutes. Reduce heat to medium-low when olive oil begins to bubble. Cook and stir until garlic is golden brown, about another 5 minutes. Remove from heat.
Stir red pepper flakes, black pepper, and salt into the pasta. Pour in olive oil and garlic, and sprinkle on Italian parsley and half of the Parmigiano-Reggiano cheese; stir until combined.
Serve pasta topped with the remaining Parmigiano-Reggiano cheese.