Special equipment: an electric multi-cooker or other electric pressure cooker
Mix together the brown sugar, smoked paprika, chili powder, black pepper, dry mustard, granulated garlic, cayenne and 2 teaspoons salt in a small bowl. Rub the pork all over with the spice mixture.
Heat an electric multi-cooker on the browning or sear function. Add the olive oil. When hot, add the pork and brown on all sides, about 6 minutes. Pour in the apple juice and 1/4 cup of the cider vinegar. Cover, lock the lid, and set the multi-cooker to High Pressure; cook for 40 to 45 minutes. Release the steam according to manufacturer's instructions.
Remove the pork to a rimmed baking sheet or large plate and tent with foil; let cool slightly. Skim off any excess fat that has risen to the top of the braising liquid, then stir in the remaining 1/4 cup cider vinegar and the ketchup. Pass some of the sauce through a strainer into a serving vessel or cruet.
With two forks, pull the pork into shreds, discarding any large pieces of fat. Add the pork back to the remaining sauce in the multi-cooker and keep warm.
Serve the pulled pork on rolls, topped with Cider Vinegar Slaw. Pass the extra sauce on the side.
Whisk together the vinegar, ketchup, sugar, celery seeds, hot sauce and 1 1/2 teaspoons salt in a large bowl until the sugar dissolves. Whisk in the oil to make a smooth dressing.
Add the cabbage and red onions to the dressing, and toss well. Cover and let stand at room temperature for about 20 minutes to wilt the cabbage slightly. (The slaw can also be made several hours ahead and refrigerated.)
Sprinkle the slaw with the parsley; toss and serve.