Preheat a grill to 500 degrees F and lightly oil the grates.
In a small saucepan combine the butter, lemon juice, olive oil, parsley, 1 teaspoon of salt, and 1/2 teaspoon pepper. Gently melt over a low flame or heat.
Brush the skin sides of the fillets with the olive oil and immediately lay the fillets skin side down on the prepared grill. Season the flesh side of the fillets lightly with additional salt, and pepper. Close the lid and cook for 15 minutes. During the last 2 minutes of cooking, baste the fillets with the butter mixture.
Once cooked, remove the fillets to a serving platter and baste for a final time with the butter mixture and serve.