Rinse the fish well and cut it into three-to-four-inch pieces. Place them in a deep-sided pan and fill it with cold water just to cover the fish. Change the water every hour or two, draining and refilling the bowl at least three times. Then cover and refrigerate overnight (or about 12 hours).
In the morning, remove the fish from the pan and throw out the soaking water. In a large pot or kettle, bring enough water to cover the fish to a simmer, place the fillets in it, and simmer gently — do not let the water boil. After 20 minutes, remove the fish from the water and set aside. Mash slightly.
While the fish is cooking, prepare the parsnips. Peel them, remove any woody centers, and steam in a double boiler over boiling water until soft, about 15 minutes. Purée the parsnips in a food processor with the butter, cream or milk, and salt.
Serve the fish with the parsnips.
