Preheat the oven to 375 degrees. Toast walnuts until they are crisp and fragrant, about 7 minutes.
In a food processor combine walnuts and flaxseeds and whirl until the seeds are crushed and somewhat ground, about 1 minute. Add the parsley, garlic, olive oil, Parmesan, 1/2 cup water, salt, and pepper. Puree until smooth and thick.
In a large pot of boiling salted water, cook 1 pound of linguine according to package instructions. Reserve 3/4 cup pasta cooking water. Drain. Spoon half the pesto into the pasta pot, and add the cooked linguine and the reserved pasta cooking water; toss well to coat. Season with salt and serve with additional Parmesan if desired.