Peel the parsnips and potatoes and cut them into ½-inch chunks. Bring a large pot of salted water to a boil over high heat. Add the parsnips and potatoes and cook until tender, 8 to 10 minutes.

While the vegetables are boiling, combine the cream, shallots, and chives in a small saucepan and bring to a simmer; cook 1 minute, then adjust the heat to keep them warm but not simmering.

Drain the parsnips and potatoes in a sieve and shake them dry. Rest a food mill on the edges of the warm, dry pot the vegetables were boiled in. Pass the vegetables through the mill directly into the pot. (If you don’t have a food mill, push the soft vegetables through a coarse sieve or mash them well with a fork. Don’t use a food processor or blender, as it will give the potatoes a gluey texture.)

Using a wooden spoon, beat in the butter and warm cream mixture. Season well with salt and pepper. Serve immediately.