Preheat the grill to medium-low heat. Brush the grates with cooking oil to help prevent sticking.
Pat the salmon filets dry with a paper towel. Drizzle with olive oil and rub to coat them evenly. Season with salt and pepper on both sides.
Once the grill is hot, place the salmon filets directly on the grill grates, flesh side down.
Grill the salmon for about 4-5 minutes on the first side, then carefully flip the filets with a spatula.
Continue grilling the salmon for an additional 3-4 minutes, or until the fish is cooked to your desired level of doneness. The internal temperature can range from 135°F(55°C) to 145°F(65°C).
While the salmon is grilling, prepare the lemon butter sauce. In a small saucepan, melt the butter over medium heat. Add the minced garlic and cook for 1-2 minutes until fragrant.
Remove the saucepan from the heat and stir in the lemon zest, and chopped parsley. Season the lemon butter sauce with a pinch of salt, if needed.
Once the salmon is cooked, transfer the filets to a serving platter or individual plates.
Drizzle the salmon with the sauce, spooning it over the top of each filet.