Tie thyme sprigs with twine or in a cheesecloth bag. Combine milk, garlic, and thyme in a small saucepan. Bring to a simmer over medium heat. Reduce heat to low and simmer gently until garlic is soft, 5 to 10 minutes. Discard thyme.
Transfer the milk mixture to a blender and purée until smooth. Return to the pan and stir in scallions. Cover and set aside.
Bring 1 inch salted water to a boil in a large pot.
Add cabbage, cover, and cook until very tender, 8 to 10 minutes. Drain well.
Meanwhile, place potatoes in a large saucepan and cover with cold salted water. Bring to a boil, reduce heat to medium and cook, covered, until tender, 10 to 15 minutes. Drain the potatoes and return to the pan. Mash with a potato masher or hand-held electric mixer.
Gradually add the milk mixture to the potatoes, stirring until smooth. Stir in cabbage and season with salt and pepper. Transfer to a warmed bowl and garnish with parsley.
Make Ahead Tip: The potatoes can be prepared up to 1 hour ahead and kept warm, covered, over a larger pan of barely simmering water.