Preheat the oven to 325 degrees F. Let the pork sit covered at room temperature for 30 minutes.
Pat the pork dry with a paper towel. Sprinkle the pork generously with salt and pepper.
Heat a large Dutch oven or heavy-bottomed pot over medium-high heat with the oil, 2 turns of the pan. Place the pork in the pot and evenly brown the meat all over, about 10 to 12 minutes time in total. Remove the pork and reserve.
Turn heat down to medium, add 3 tablespoons of the butter to the pot. When it foams, add the caraway, fennel and mustard seeds and stir for 1 minute. Add in the onions, apples, thyme, nutmeg and some salt and pepper. Cook to begin to soften, 5 minutes. Then deglaze the pot with the vinegar. Stir in the honey and beer, scraping up the bits stuck to the bottom of the pot. Add in the chicken stock and then the reserved pork, and bring to a simmer. Cover and place the pot in the oven until the meat is very tender, 2 1/2 to 3 hours, turning the meat halfway through cooking.
Remove the pork from the pot, place on a plate and tent with foil to keep warm. When cool enough to handle, pull the meat apart with two forks. Return the meat to the braising liquid. The apples and onions will have almost completely melted into a chunky sauce. Cool and store in the refrigerator for a make-ahead meal.
Place the potatoes and the parsnips in a pot and cover with water. Bring to a boil, salt the water and cook until tender. Drain and return to the hot pot. Mash the potatoes with the milk and the remaining 3 tablespoons butter. Then season with salt and pepper, and add the cheese and mustard. Taste to adjust the seasonings, cool and store for a make-ahead meal.
To serve, reheat the pork, covered over medium heat, stirring occasionally. Reheat the mashed potatoes and parsnips over medium heat covered with a little stock or milk to loosen. Mashed potatoes can be kept warm for hours by placing them in a covered saucepot inside of a larger pot filled with an inch of water at a low simmer.
Serve the pork alongside or on top of the potatoes, and sprinkle chopped parsley on everything.