For the rub and pork: In a small bowl, mix the paprika, smoked salt, onion powder and a few grinds of pepper. Rub the pork on all sides with the blend. Let the pork rest at room temperature for 1 hour.
In a large Dutch oven on high heat, add the butter and olive oil. Once it begins to swirl, add the pork and allow it to sit and sear to a deep golden brown, then rotate and repeat on all sides, 3 to 5 minutes per side. Don't forget the ends!
For the pot: Add the apricots around the edges of the pork and cook while moving around for a minute. Add the thyme, garlic, rosemary and enough coconut milk to reach three-quarters of the way up the sides of the loin. Bring to a boil, then lower to a very low simmer and cover. Cook for 1 hour, then flip the loin and cook until tender, another hour.
Remove the loin to a plate and cover with aluminum foil for 10 minutes before slicing into 1-inch-thick slices.
For the gravy: Remove the herbs from the pot and raise the heat to reduce the coconut milk to a gravy consistency. Taste and season with salt if needed. Stir in the parsley and lime juice, and spoon over the pork slices.