Preheat oven to 400°F (200°C).
Pat the salmon fillets dry with paper towels and season both sides with salt and black pepper.
In a large oven-safe skillet, heat 2 tablespoons of butter and 1 tablespoon of olive oil over medium heat.
Add the salmon fillets skin-side down and cook for 4 minutes without moving them to get a crispy skin.
Flip the salmon fillets and add the minced garlic to the skillet. Cook for 1 minute until fragrant.
Add the trimmed asparagus around the salmon in the skillet. Season asparagus with salt and black pepper.
Drizzle the lemon juice and sprinkle the lemon zest over the salmon and asparagus.
Add the remaining 2 tablespoons of butter in small pieces over the salmon and asparagus.
Transfer the skillet to the preheated oven and roast for 8-10 minutes, or until the salmon is cooked through and asparagus is tender.
Remove from oven and sprinkle fresh chopped parsley over the salmon and asparagus before serving.