Put the parsnips and potatoes in a large saucepan and cover with about 2 inches of water. Season the water generously with salt - TASTE IT to make sure the water is seasoned appropriately.
Bring the water to a boil and reduce to a simmer. Cook the parsnips and potatoes until they are fork tender, about 30 minutes.
In a small saucepan, bring the heavy cream to a boil. Turn off the heat and reserve.
Drain the parsnips and potatoes and pass through a food mill.
Toss in a couple of pats of cold butter and about 1/2 of the hot heavy cream. Stir vigorously to combine. Repeat the process with the remaining butter and heavy cream. Taste and adjust the seasoning if needed.