Heat oil in a heavy large deep pan or dutch oven over medium high heat.
Season the ribs with salt and pepper and sear until browned on all sides. About 30 seconds on each side. Remove them to a plate.
Leaving the brown bits in the pan, add the vegetables and garlic. Stir them, and continue to cook until just softened. Now, Add the tomato paste, and stir for another minute.
Pour in the red wine, and the beef stock. Bring the mixture to a soft boil, and add the seared short ribs back to the pan.
Add the fresh thyme and dried bay leaves to the pan.
Place the lid on the pan, reduce the heat to a low simmer; you should see minimal movement in the liquid. Place the pan in a preheated 300 degrees F. oven if you prefer the oven method. Allow the meat to braise until very tender.
While the short ribs braise, cook the pasta according to the directions on the package.
Remove the beef to a plate. Strain the liquid in the pan. Wipe out the pan, and return half of the strained liquid to the pan. Allow the liquid to reduce by half.
Place the short ribs over the pasta, drizzle on a bit of the reduced braising liquid, and garnish with fresh parsley.