Preheat the oven to 375 degrees F (190 degrees C).

Pat pork shoulder with kitchen towels or paper towels to remove any excess moisture. Season all sides evenly with truffle salt and white pepper.

Heat olive oil and truffle oil in a large Dutch oven over medium-high heat. Add pork shoulder to the hot oil and brown thoroughly on all sides, 3 to 5 minutes per side. Transfer meat to a plate.

Lower heat to medium and melt 2 tablespoons truffle butter in the Dutch oven. Add potatoes and season with truffle salt and white pepper. Saute until potatoes are golden brown but not cooked through, about 10 minutes. Transfer potatoes to a large bowl.

Add onions, parsnips, and celery to the pot; cook until softened, adding more olive oil to prevent burning if necessary, about 5 minutes. Add mushrooms and cook until softened, about 3 minutes. Add garlic and cook until fragrant, 1 to 2 minutes. Transfer vegetables to a medium bowl.

Return heat to medium-high and pour in wine; bring to a boil while scraping the browned bits of food off the bottom of the pan with a wooden spoon. Add pork back to the pot and surround it with browned potatoes. Add vegetable broth until liquid covers the potatoes.

Cover and bake in the preheated oven for 45 minutes. Flip the pork and bake an additional 45 minutes. Transfer 2 to 3 cups cooking juices to a bowl and set aside for sauce. Baste the pork in the pot with the remaining cooking juices. Add onion-mushroom mixture, cover, and bake until vegetables are tender and pork is done, 15 to 20 minutes longer. An instant-read thermometer inserted into the center should read at least 145 degrees F (63 degrees C).

Meanwhile, melt remaining 2 tablespoons truffle butter in a saucepan over medium-low heat. Whisk in flour and stir until the mixture becomes paste-like and light golden brown, about 5 minutes. Gradually whisk the reserved cooking juices into the flour mixture, and bring to a simmer over medium heat. Cook and stir until the mixture is thick and smooth, 10 to 15 minutes.

Serve pork with vegetables, potatoes, and sauce. Sprinkle with fresh parsley.