In a heavy 12-inch skillet, melt 3 Tbs. of the butter over medium-high heat. Add the mushrooms and cook, stirring only occasionally, until well browned, 5 to 7 minutes. Add the garlic and cook,stirring, until golden and fragrant, about 45 seconds. Add the vermouth, 3 Tbs. of the parsley, the vinegar, salt, and pepper and boil until the liquid is reduced by half, about 2 minutes.
Remove the pan from the heat, lightly season the cod with salt and pepper and add it to the pan (tuck under the tails if necessary to even out the thickness), nestling the fillets into the mushrooms and spooning some of the mushrooms on top.
Bring the mixture to a gentle simmer over medium heat, cover the pan, reduce to medium low, and simmer until just cooked through(use the tip of a paring knife to check), 7 to 12 minutes, depending on thickness. With a slotted spatula, transfer the fish to serving plates or a platter. Over low heat, whisk the remaining 1 Tbs. butter into the sauce. Spoon the sauce over the fish and serve sprinkled with the remaining 1/2 Tbs. parsley.
