Heat olive oil in a large skillet over medium-low heat and cook onion until soft and translucent, about 5 minutes. Add napa cabbage, sprinkle with sugar, and mix well. Cook and stir over low heat, 3 to 5 minutes. Season with ginger, cumin, salt, and pepper and cook for 1 more minute.
Pour sake and broth into the skillet with the cabbage and bring to a boil. Reduce heat, cover, and simmer for 10 minutes. Stir in balsamic vinegar and adjust seasoning to taste. Mix in parsley.