Pat salmon dry and sprinkle on one side with paprika, salt, and pepper. (If the salmon has skin, season the skinless side.)
Whisk together honey, soy sauce, vinegar, and garlic in a small bowl or liquid measuring cup; set aside.
Warm olive oil in a large skillet over medium-high heat. When hot, add salmon to the pan, with the unseasoned side down, and cook for 3-4 minutes. Test the salmon; when the bottoms release fairly easily from the pan, it’s ready to flip. Turn each filet over, and cook for an additional 1-2 minutes.
When salmon is nearly done, pour sauce evenly on top. Let the sauce bubble for about 1 minute, while drizzling spoonfuls from the bottom of the skillet over the salmon. Remove from the heat, sprinkle with parsley, if desired, and serve right away.
If possible, it’s helpful to remove salmon from the fridge and let it rest at room temperature for 15-20 minutes prior to cooking. This helps it cook more evenly and release more easily from the pan after cooking on the first side.
There are several ways to tell when salmon is done: the color will change from translucent, raw red, to a more opaque pink, and it will flake easily when pressed gently with a fork or your finger—this means that it separates easily along the white lines that criss-cross the filet, which are strips of fat. The FDA recommends cooking salmon to an internal temperature of 145 degrees F.
Keep in mind that the salmon may continue to cook briefly after you remove it from the heat, particularly if using a cast-iron skillet or other cookware that holds heat very well.