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Bring a large pot of salted water to a boil. Meanwhile, heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the kielbasa and cook, undisturbed, until crisp on the bottom, about 2 minutes. Toss and continue to cook until crisp and browned on the other side, about 2 more minutes. Remove with a slotted spoon to a plate. Reserve the skillet.
Add the remaining 2 tablespoons olive oil to the skillet, then add the cabbage, onion and a big pinch of salt; toss to coat. Cook, stirring occasionally, until very tender and golden brown, 20 to 24 minutes. Add the garlic, thyme and red pepper flakes and cook until softened, about 1 minute. Add the vinegar and cook until the liquid is absorbed, 1 more minute.
During the last 8 minutes of cooking the cabbage, add the spaghetti to the boiling water and cook as the label directs for al dente. Reserve 1 cup cooking water, then drain. Add the pasta to the skillet along with the cheese, ½ cup of the reserved cooking water and the kielbasa; toss to combine, adding more cooking water to loosen, if needed.
Season with salt and stir in the parsley. Divide among plates.