Preheat the oven to 200 degrees F.
Generously coat a large saute pan with olive oil and bring to a high heat. Season 2 of the fillets with salt. Dredge the fillets in flour and place them immediately in the pan with the hot oil. Do not flour the fish ahead of time or it will be gummy and mushy on the outside rather than crispy.
When the fish has turned from translucent to opaque, about 2/3 of the way through the cooking, turn the fish over and cook on the other side. Remove the fish from the pan and reserve on a sheet tray in the oven. Repeat the process with the remaining 2 fish fillets. Keep the cooked fish fillets warm in the oven while you make the sauce.
Remove any excess oil from the pan. Add the butter and thyme leaves. Shake the pan frequently to prevent scorching. When the butter has melted and is very bubbly, add in the lemon juice and whisk to combine. As the butter becomes bubbly again and starts to turn a nutty brown, season with salt and whisk in the chopped parsley. Taste to make sure it is delicious.
Remove the fish fillets from the oven and plate them, spooning the sauce over the fish.