1. Put parsnips and potatoes in a medium pot with enough water to cover by 1 in. Bring to a boil over high heat, then cover, reduce heat, and simmer until tender, 8 to 12 minutes. Drain vegetables and return to pot. Cook vegetables, stirring, over low heat 1 to 2 minutes to cook off excess moisture.
2. Mash vegetables in pot or beat in bowl of a stand mixer until very smooth. Stir in butter, half-and-half, and salt and pepper to taste.
*For a nondairy kosher version, use extra-virgin olive oil in place of the butter and hot chicken broth in place of the half-and-half.
Make ahead: Chill airtight up to 1 day; microwave until hot, stirring occasionally.
Note: Nutritional analysis is per 1/2-cup serving.