Cover shredded cabbage in water in a large pot and bring to a boil. Cover pot, reduce heat and simmer for about 10 minutes.
Drain the cabbage, reserving the water from the pot. Set aside the cabbage for now. Add the cubed potatoes to the water in the pot. Add enough water to cover the potatoes. Bring to a boil and boil uncovered for about 15-20 minutes, or until potatoes are fork tender.
While the potatoes are boiling, combine the milk, 4 tbsp of butter, green onions, salt, and pepper in a small saucepan. Warm the ingredients to incorporate them well, stirring occasionally.
When the potatoes are done, drain them and return them to the pot. Add the remaining 4 tbsp of butter and the milk mixture, then mash until smooth and well incorporated. Fold the boiled cabbage into the potatoes.
Garnish with crumbled bacon pieces and chopped parsley. Serve with extra butter, salt, and pepper.