Cook pasta according to package directions. Drain, reserve 1 cup cooking liquid.
While the pasta cooks, heat walnut oil in a large skillet over medium-high. Add garlic and cook, stirring often, until golden, 3 to 4 minutes. Stir in parsley; remove from heat. Let oil mixture cool for 2 minutes.
Add ½ cup reserved cooking liquid to oil mixture, and return to medium-high heat. Cook until sauce has reduced by half, about 5 minutes. Stir in cooked pasta. Add ¼ cup reserved cooking liquid and cook, sirring constantly, until sauce has thickened about 2 minutes. Stir in remaining ¼ cup cooking liquid as needed to reach desired consistency. Season with salt, top with cheese and black pepper.