Bring a large pot of water to boil over high heat.
Add potatoes and cook for 12 minutes. Add parsnips and cook for an additional 8-10 minutes until both potatoes and parsnips are tender. Drain well.
Mash potatoes and parsnips together with coconut milk and salt.
Meanwhile, melt butter in a large pan. Add onion and sauté for 5-7 minutes, stirring occasionally, until very soft. Add cabbage to the onions and cook, stirring often for 5 minutes more or until very soft.
Stir cabbage mixture and cheese into hot potatoes and parsnips and season with pepper. Top with additional shredded cheese and melted butter if desired.