Pat the salmon fillets dry with paper towels and season both sides with salt and black pepper.
Heat 2 tablespoons of butter and 1 tablespoon of olive oil in a large skillet over medium-high heat.
Place the salmon fillets skin-side down in the skillet and cook for 4-5 minutes without moving until the skin is crispy.
Flip the salmon fillets and add minced garlic to the skillet. Cook for another 3-4 minutes, spooning the melted butter and garlic over the salmon as it cooks.
Remove the salmon from the skillet and set aside, keeping warm.
In the same skillet, add the remaining 2 tablespoons of butter.
Add the trimmed asparagus to the skillet and sauté for 4-5 minutes until tender-crisp.
Add the lemon juice and lemon zest to the asparagus, stirring to coat evenly. Season with salt and black pepper to taste.
Serve the salmon fillets on plates alongside the lemon asparagus.
Garnish with chopped fresh parsley before serving.