Put the parsnips in a 4-quart saucepan and fill two-thirds full with cool water. Add 1 tsp. salt and bring to a boil. Turn the heat down and cook at a gentle boil until the parsnips can be easily pierced with a fork, about 7 minutes.
Drain well in a colander, letting the steam rise off for a few minutes. Meanwhile heat the butter, cream, mustard, honey, 1 tsp. salt, and 1/2 tsp. white pepper in the saucepan over low heat until the butter is melted. For a rustic texture, return the parsnips to the pan and mash with a potato masher to the consistency you like.
For a smooth texture, purée the parsnips in a food processor until smooth, and then add them to the pan, stirring well to combine. Season to taste with more salt and pepper, and a pat of butter if you like, before serving.