Put the potatoes and parsnips in a large heavy-bottomed saucepan, cover with cold water and salt it generously. Bring the potatoes to a boil over high heat; reduce heat to medium and simmer until fork tender, 20 to 25 minutes. Put the butter and half-and-half in a small saucepan over medium-low heat until the butter is melted and the mixture is hot.
Drain the potatoes and parsnips well and return them to the hot pan. Stir the vegetables in the pan to dry them out a bit. Add the hot butter mixture and season with salt and pepper. Mash with a potato masher until smooth. Transfer to a serving bowl and garnish with chopped chives.