1. Preheat oven to 450 degrees. Butter a 9-by-13-inch glass or ceramic baking dish. Sprinkle bottom of dish with salt and pepper; set aside. Finely chop 12 olives, and thinly slice remaining 4; reserve separately.

2. Slice potatoes very thinly (about 1/16 inch thick) with a mandoline or sharp knife. Arrange one-third of potatoes in buttered dish, overlapping slightly. Brush with melted butter; season with salt and pepper. Scatter one-third of garlic and half of the chopped olives on top. Repeat process to make another layer. Top with remaining potatoes; brush with melted butter, and season with salt and pepper. Set aside remaining garlic and butter.

3. Tightly cover with foil. Roast until potatoes just begin to color, 16 to 18 minutes. Remove foil; roast until edges are pale golden, about 10 minutes more. Meanwhile, skin fillets if necessary; halve crosswise. Stir together reserved garlic and melted butter in a large dish. Dip fillets in mixture; season with salt and pepper.

4. As soon as potatoes come out of oven, place fillets on top. Roast until fillets are cooked through, about 10 minutes. Scatter olive slices and parsley on top.