In a large pot, bring 4 quarts water to a boil. Add the pasta and 1 tablespoon salt, then cook, stirring occasionally, until al dente. Reserve 1 cup of the cooking water, then drain; set aside.
Meanwhile, in a 12-inch skillet over medium, toast the walnuts, stirring often, until fragrant and lightly browned, about 3 minutes. Transfer to a cutting board; reserve the skillet. Let the nuts cool slightly, then finely chop; set aside.
In the same skillet over medium-high, combine the oil and garlic; cook, stirring often, until the garlic begins to brown, about 30 seconds. Stir in the cabbage, butter, ½ teaspoon salt and ¼ teaspoon pepper. Distribute the cabbage in an even layer, then cook, stirring only once or twice, until well browned and tender-crisp, 2 to 4 minutes.
Stir the pasta into the cabbage mixture, then add half of the walnuts and ½ cup of the reserved cooking water. Reduce to medium and cook, stirring constantly and vigorously, until the pasta is lightly sauced, 1 to 3 minutes, adding more cooking water as needed to make a silky sauce.
Off heat, stir in the lemon juice and half the parsley. Taste and season with salt and pepper. Serve sprinkled with the remaining walnuts and parsley, Parmesan and generous grindings of pepper.
