Render fat from bacon in a large saucepan over medium-low heat, about 15 minutes. Add onions and carrots and cook, stirring occasionally, for 5 minutes. Sprinkle vegetables with flour and continue cooking, stirring occasionally, for 10 minutes. Add tomato paste, parsley, bay leaves, thyme, and 2 quarts (8 cups) beef stock. Simmer, skimming occasionally, over medium heat until sauce has reduced by three-quarters, about 3 hours. Strain sauce, discard solids and return to pan. Add remaining 2 cups beef stock and simmer over medium heat until sauce has reduced by half, about 2-1/2 hours, then strain.
(Demi-glace can be stored in the refrigerator in a sealed container for up to 1 week or frozen for up to 6 months.)