Heat oil in a medium pan over medium heat until shimmering. Meanwhile, very thinly slice garlic with a mandoline (you should have 1/4 cup). Add to oil, and cook, stirring constantly, until garlic is pale golden brown and crisp, about 3 minutes. Drain garlic chips in a sieve set over a large bowl (reserve oil). Spread garlic chips on paper towels; season with salt.
Bring a large pot of salted water to a boil. Cook pasta until just barely al dente. Drain. Add pasta to oil, and toss. Partially cover, and let cool at least 20 minutes and up to 1 1/2 hours.
Roughly chop 3/4 cup parsley leaves, and add them with parsley stems, lemon zest, and walnuts to pasta. Toss to combine. Dot with cheese, and season with salt and pepper. Toss gently to combine, being careful not to fully incorporate cheese. Garnish with remaining parsley leaves, and serve with garlic chips.