Trim the excess fat from the chuck roast, then cut the meat into quarters. Transfer to a 6-quart slow cooker and toss with the flour, paprika, thyme, 1/2 teaspoon salt and a few grinds of pepper. Add the onion, garlic, mushrooms and broth. Cover and cook on low until the beef is fork-tender, 7 to 8 hours.
Skim off any excess fat from the slow cooker with a ladle. Using two forks, break up the beef into bite-size pieces and keep warm.
Bring a large pot of salted water to a boil. Add the egg noodles and cook as the label directs. Drain and return to the pot; toss with the butter.
Divide the noodles among bowls. Add the sour cream to the slow cooker and stir to combine. Divide the beef mixture among the bowls and top with parsley.