Pat short ribs dry with paper towels. Season all sides generously with salt, black pepper, and smoked paprika. Lightly dust with flour, shaking off excess. Let sit at room temperature for 15-20 minutes while prepping vegetables.
Set Instant Pot to Sauté on High or More. Add olive oil and heat until shimmering. Working in batches, brown ribs for 3-4 minutes per side until deeply caramelized. Transfer browned ribs to a plate and set aside.
With pot still on Sauté, add diced onion and carrots. Cook 4-5 minutes until onion is translucent. Add minced garlic and tomato paste, stirring constantly for 1-2 minutes until fragrant and slightly darkened.
Pour in red wine, scraping the bottom thoroughly with a wooden spoon to release all browned bits. Let wine come to a simmer and cook for 2-3 minutes to reduce by about one-third.
Add beef broth, stirring to combine. Add thyme, rosemary, and bay leaves. Return browned short ribs to the pot, nestling them into the liquid and turning once to coat. Liquid should come halfway up the meat.
Turn off Sauté. Secure lid, set valve to Sealing, and select Pressure Cook or Manual on High for 45 minutes. After cooking, allow 15 minutes natural release, then carefully vent any remaining pressure.
Remove ribs to a platter and cover with foil. Discard herb sprigs and bay leaves. Skim fat from the cooking liquid. Return pot to Sauté on Medium. For thicker sauce, whisk cornstarch slurry into simmering liquid and cook 3-5 minutes until thickened. Season to taste.
Arrange short ribs on plates or a platter. Spoon rich sauce generously over the meat. Garnish with fresh parsley and serve immediately while hot.