Preheat grill to medium-high heat. Meanwhile, season the salmon with salt and pepper.
Melt 3 tablespoons of butter in a pan over medium heat along with minced garlic cloves for about 1 minute until fragrant. Add lemon zest (zest one whole lemon or half) to melted butter, stirring occasionally. Remove the grill and place salmon fillets skin side down on it.
Grill each piece of salmon for approximately 4-5 minutes until cooked through. While they are still warm (and in their own juices), drizzle with lemon butter sauce made from melted remaining butter, fresh dill and the rest of the minced garlic cloves.
Garnish each fillet liberally with chopped parsley before serving.