How to Make It
In a large saucepan, cover the parsnips with cold water. Add a large pinch of salt and bring to a boil. Simmer the parsnips over moderate heat until very tender, about 25 minutes. Drain the parsnips well, reserving 1/2 cup of their cooking liquid.
In a food processor, pulse the parsnips until coarsely chopped. Add the reserved cooking liquid and 2 tablespoons of the butter and season with salt; process until smooth. Return the parsnip puree to the saucepan and keep warm.
Put the flour in a shallow bowl. Season the fish fillets with salt and pepper, then dredge them in the flour; tap to remove any excess flour. Melt 2 tablespoons of butter in each of 2 large nonstick skillets. Add 4 fillets to each skillet, skin side down. Cook over moderately high heat, turning once, until golden brown and cooked through, about 8 minutes.
Spoon the parsnip puree onto warmed plates. Top with the fish fillets, skin side up, and serve with the Balsamic-Glazed Red Onions.