Season the salmon with salt, pepper, garlic powder, and dried thyme.
Heat the olive oil and butter over medium-high heat in a large nonstick skillet, swirling to coat.
When the butter starts foaming, add the salmon fillets to the pan, skin-side-down. If using, add the lemon slices too.
Fry the salmon, undisturbed, until the skin is crisp and browned, about 4 minutes.
Flip the fish and the lemon slices over, lower the heat to medium, and continue cooking until the fish is cooked through, 2-3 more minutes.
Transfer the salmon to a plate and baste it with the pan juices. Top each fillet with a lemon slice, sprinkle with chopped parsley, and serve.
If your salmon is frozen, I recommend defrosting the salmon before using it in this recipe. It's fine to use frozen salmon when baking it in the oven. But when pan-frying, I feel that it should be thawed to ensure it is evenly cooked.Don't forget to drizzle (or brush) the salmon with the pan juices after you've arranged it on plates. The pan sauce is amazing and adds tons of flavor to the dish.