In a large saucepan or skillet, melt 3 tablespoons of butter over moderate heat.
Add the mushrooms and simmer for 5 to 7 minutes, until tender. Stir occasionally.
Add the garlic and simmer until fragrant, approx. 30 seconds.
Slowly add the beer, the chopped parsley, the balsamic vinegar, salt, and pepper.
Bring to a boil and cook until liquid is reduced by half, approx. 2 minutes.
Remove the pan from the heat.
Lightly season the cod with salt and pepper.
Place the fillets into the pan.
Spoon some of the sauce and mushrooms on top of the fillets.
Place the pan back onto the stove over moderate heat and bring to a simmer.
Cover, reduce heat to medium-low, and simmer until cooked through (internal temp of 145 degrees F), approx. 7 to 10 minutes.
With a slotted spoon or spatula, carefully transfer the fish to serving plates.
Over low heat, whisk the remaining butter into the sauce until melted.
Spoon the sauce and mushrooms over the fish and garnish with a bit of fresh chopped parsley.