Pat the short ribs dry, then season all over with salt and pepper.
Heat the olive oil in a heavy pot over medium-high. Sear ribs on all sides until deeply browned. Transfer to a plate.
Lower heat to medium. Cook onion, celery, and carrots in the rendered fat until softened, about 6–8 minutes.
Stir in garlic and tomato paste. Cook until fragrant and darkened slightly, about 1 minute.
Sprinkle in the flour and cook, stirring, for 1 minute to coat the vegetables.
Pour in the ale, scraping up browned bits. Simmer 2–3 minutes to reduce slightly.
Add beef broth, Worcestershire, thyme, and rosemary. Return the ribs and any juices to the pot and bring to a gentle simmer.
Cover and simmer on low until the ribs are nearly tender, about 1 hour 30 minutes, adjusting heat to maintain a gentle bubble.
Stir in potatoes and mushrooms. Cover and cook until ribs are very tender and vegetables are soft, 35–45 minutes more.
Lift out ribs and discard excess surface fat from the stew. Shred meat from the bones, return to pot, and stir in balsamic. Adjust salt and pepper. Garnish with parsley and serve.