In a large pot, bring 4 quarts water to a boil. Stir in the pasta and 1 tablespoon salt, then cook, stirring occasionally, until al dente. Reserve about ½ cup of the cooking water, then drain in a colander and return to the pot.
While the pasta cooks, in a food processor, combine the walnuts, Parmesan, ¼ teaspoon salt and ½ teaspoon pepper. Process until the mixture resembles coarse sand, 10 to 20 seconds. Add the parsley, oil and capers; process, scraping the bowl as needed, until almost completely smooth, about another 30 to 40 seconds.
Add the pesto to the pasta in the pot along with ¼ cup of the reserved pasta water, then toss; add more reserved pasta water as needed so the pesto coats the noodles. Taste and season with salt and pepper. Serve sprinkled with grated Parmesan.
