Cut 3-in. lengths from slender ends of carrots, parsnips, and celery. Halve them lengthwise, then halve again if large; you should have 12 to 14 sticks of each vegetable. Cut remaining vegetables into 1/2-in. dice; separate and set aside diced vegetables and sticks. Preheat oven to 375°.
Sprinkle short ribs all over with 1/4 tsp. each salt and pepper. Heat olive oil in a 6- to 8-qt. pot over medium-high heat. Working in 3 batches, brown ribs in oil on both sides, 5 to 10 minutes per batch. Transfer as browned to a 12- by 17-in. roasting pan. Pour fat from pot and reserve; return 2 tbsp. to pot.
Add diced vegetables and onions to pot and cook, stirring often, until lightly browned, 5 to 10 minutes. Spoon over ribs. Pour reserved fat into pot. Add vegetable sticks and cook, stirring often, until golden, 5 to 6 minutes. Spoon into an 8-in. square pan and sprinkle with a little chopped parsley, salt, and pepper. Cover tightly with foil; chill until roasting (up to 1 day).
Pour wine, tomatoes, 1/3 cup parsley, and 3/4 tsp. each salt and pepper into empty pot. Bring to a boil over high heat, then pour over ribs. Turn ribs in liquid to coat. Scatter garlic cloves and rosemary on top. Cover tightly with 2 sheets foil.
Braise rib mixture in oven, turning meat every hour, until meat is nearly fork-tender, about 2 1/4 hours. Skim and discard fat from juices.
Meanwhile, fill a medium bowl with water and add lemon juice. Trim stems, outer leaves, and tips from artichokes, leaving just the tender yellow-green leaves, and trim bases to neaten; drop each into lemon water immediately after trimming to prevent browning (keep in water up to 2 hours).
Drain artichokes and push into liquid around ribs; cover tightly. Cook rib mixture and pan with vegetable sticks until ingredients are very tender, 25 to 30 minutes.
Put a large platter in oven to warm. Discard rosemary stems from roasting pan. If mixture is very juicy, transfer ribs and artichokes to a large bowl and cover; set pan over 2 burners on high heat and boil, stirring often, until most of liquid evaporates, 5 to 7 minutes.
Spoon diced vegetables and juices from roasting pan onto platter. Arrange ribs, artichokes, and vegetable sticks on top. Sprinkle with chopped parsley and garnish with parsley sprigs. Serve with Horseradish Beet Sauce.
*Ask a butcher to cut short ribs into twelve 4-in.-long pieces.
Make ahead: Make through step 5, then cool and chill airtight 1 day. Spoon off fat from juices and discard, then reheat rib mixture, covered, in a 375° oven for 45 minutes; continue with step 6, adding about 1 cup water if pan is low on juices.