In a large skillet, heat oil over medium-high. Season short ribs with salt and pepper. In batches, cook until browned on all sides, about 10 minutes total. Transfer ribs to a 5- to 6-quart slow cooker. Pour off all but 1 teaspoon fat from skillet, and add onion and carrot. Season with salt and pepper and cook, stirring, until vegetables are soft, 3 minutes. Add red wine, stirring and scraping up any browned bits from skillet with a wooden spoon.
Transfer vegetables and liquid to slow cooker and add tomatoes, breaking them up with your hand. Add herb sprigs; season to taste with salt and pepper. Cover and cook on high (300F) until meat is fork-tender, 6 -8 hours. Check after 6 hours and continue to cook if needed. The length of time may vary depending on the wattage of your slow cooker.
Once the meat is fork tender, remove the herb sprigs. If the sauce is too thin, transfer the meat to a bowl and cover with foil to keep warm. Pour the sauce into a medium saucepan and cook uncovered until it reduced and thickened. Add the meat back to the sauce.
Serve over hot pappardelle pasta (or other wide noodles) and garnish with chopped fresh parsley.