Season the top of fish fillets with Italian herb seasoning and salt (generously). Fish fillets will have skins on the bottom - no need to season the skins.
In a large skillet (large enough to fit 2 fish fillets), heat 2 tablespoons of olive oil on medium heat until heated but not smoking. To the hot skillet with olive oil, add fish fillets skin side up - flesh side down. Cook the flesh side of the fish for about 3-5 minutes on medium heat, making sure the oil does not smoke, until lightly browned.
Flip the fillets over to the other side, skin side down (add more oil, if needed). Cook for another 2-4 minutes on medium heat (to prevent oil from burning).
Remove the skillet from heat, close with the lid, and let the fish sit for 5-10 minutes, covered, in the skillet, until flaky and cooked through completely.
After the fish is cooked through, off heat, using spatula, carefully remove fillets to the plate, separating the fish from the skin. Carefully remove or scrape the fish skin off the bottom of the pan, making sure to leave all the cooking oils in the pan.
Add diced garlic, lemon juice, and white wine to the same pan with oil. Cook on medium-low heat for about 1 minute, until garlic softens a bit. Remove from heat. Add 1 tablespoon of chopped parsley, and 2 tablespoons of butter, off heat, to the sauce, stirring, until the butter melts and forms a creamy mixture.
Add fish to the pan, spoon sauce over the fish, top the fish and sauce with the remaining 1 tablespoon of parsley, and serve.