Preheat the oven to 400°F (200°C).
Place the trimmed asparagus on a baking sheet. Drizzle with 1 tablespoon olive oil and season with salt and black pepper. Toss to coat evenly and spread in a single layer.
Roast the asparagus in the preheated oven for 10 minutes.
While the asparagus is roasting, season the salmon fillets with salt and black pepper on both sides.
In a large skillet over medium heat, melt 2 tablespoons of the butter.
Add the minced garlic to the melted butter and sauté for 1-2 minutes until fragrant, being careful not to burn the garlic.
Add the salmon fillets to the skillet, skin side down. Cook for 4-5 minutes without moving them to allow the skin to crisp.
Flip the salmon fillets and cook for another 3-4 minutes until the salmon is cooked through and flakes easily with a fork.
During the last 2 minutes of cooking, add the remaining 2 tablespoons of butter, lemon zest, and lemon juice to the skillet. Spoon the garlic butter sauce over the salmon fillets continuously.
Remove the asparagus from the oven and transfer to serving plates.
Place the cooked salmon fillets on top of the asparagus and spoon any remaining garlic butter sauce from the skillet over the salmon and asparagus.
Garnish with chopped fresh parsley and serve immediately.