Place a large skillet over medium heat, add oil and wait for it to start sizzling. It is important for the skillet and oil to be hot, otherwise, the salmon may stick to the pan.
Season salmon fillets with salt and pepper and add them skin side down to the skillet. Do not over-crowd, if needed cook in batches.
Cook for 5-6 minutes without moving the fish, using a spatula, gently flip it over and cook for 3 more minutes; remove from skillet and set aside on a plate. Cover with foil to keep it warm.
In the same skillet prepare the sauce.
First, add the butter and melt it. Add the flour, and cook it for 1 minute while stirring.
Add the cream or milk, little by little, while stirring all the time until fully combined.
Next, add the rest of the ingredients with the exception of grated parmesan.
Bring sauce to a simmer, reduce heat to low and stir in grated cheese. Stir until fully melted.
Once the sauce is done, add back the cooked fish fillets and simmer on low in the sauce for 3-4 minutes.
Serve garnished with fresh parsley and lemon wedges on the side.