Heat the olive oil in a large skillet over medium heat. Add the onion and cook about 8-10 minutes, until golden and soft. Add the garlic and cook another minute, then stir in the cabbage and season with salt and pepper to taste. Cover and cook, stirring periodically, about 5 minutes.
Add the broth, stir, cover, and cook until the cabbage is tender, another 10 minutes or so. Uncover and cook until the liquid evaporates. Stir in the parsley and balsamic vinegar. Taste and add more salt or pepper if needed.