Heat olive oil in a large Dutch oven over medium-high heat. Season short ribs generously with salt and pepper, then brown on all sides for about 10 minutes until deeply caramelized. Remove and set aside.

Add chopped onion, carrots, and celery to the same pot. Sauté for 5-7 minutes until vegetables soften and onions become translucent. Add minced garlic and cook for another minute until fragrant.

Stir in tomato paste and cook for 2-3 minutes until it darkens slightly and becomes aromatic. Add crushed tomatoes and beef broth, scraping up any browned bits from the bottom of the pot.

Add thyme, rosemary, and bay leaf. Return browned short ribs to the pot, nestling them into the sauce so they're mostly covered. Cover with lid and reduce heat to low.

Braise for 3-4 hours, checking occasionally, until meat is fork-tender and falling off the bone. Remove ribs, shred the meat with two forks, discard bones, and return shredded meat to the sauce.

Meanwhile, boil peeled potatoes in salted water for 15-20 minutes until tender. Drain, then mash with butter, warm milk, and Parmesan cheese until smooth and creamy. Season with salt and pepper.

Serve generous portions of Parmesan mashed potatoes topped with short rib ragu. Garnish with fresh parsley if desired.