How to Make It
Stir together broth, tomato paste, parsley stems, 1 teaspoon salt, and pepper in a 6-quart slow cooker. Add short ribs, onions, carrots, and 2 cloves garlic; nest ribs into vegetables. Cover and cook on high until meat is very tender, 5 to 6 hours. Reduce to warm and remove lid.
Meanwhile, place potatoes, rosemary, remaining 2 cloves garlic, and 2 tablespoons salt in a stockpot; add water to cover. Bring to a boil over high. Reduce heat to medium-low and simmer until potatoes are very tender, about 15 minutes. Remove from heat. Remove rosemary and garlic, then drain.
Reserving cooking liquid, remove short ribs and vegetables from slow cooker. Skim fat from liquid, then transfer to a blender; add remaining ￼¼ teaspoon salt. With center piece of blender lid removed, secure lid on blender. Place a clean dish towel over lid opening. Puree on medium-high until smooth.
Serve potatoes and short ribs with vegetables and sauce. Garnish with parsley leaves and pepper.