Put the potatoes in a large pot and cover with cold water by 1 inch; season generously with salt. Bring to a boil, then reduce the heat to a simmer and cook until the potatoes are tender, 20 to 25 minutes.
Meanwhile, combine the milk, horseradish and a pinch of salt in a small saucepan. Bring to a simmer, then reduce the heat to low and cook, stirring occasionally, until the horseradish is tender and infuses the milk, about 10 minutes (it’s OK if the mixture looks curdled). Cover to keep warm.
Drain the potatoes well, then return to the pot over low heat. Cook until the excess liquid evaporates, 1 to 2 minutes. Add the sour cream and butter and smash with a potato masher until the butter is melted and the potatoes are mostly smooth but with some chunks. Mash in the milk mixture and parsley until creamy; season with salt and pepper. If the potatoes still seem thick, add more warm milk as needed. Transfer the potatoes to a serving dish.