Birds really shouldn't have bread, period. It's a nutrient deficient filler that does nothing but make them feel full for a moment so they don't bother going after seed, and if they get into the habit of consuming bread they prefer it because of the ease with which it's digested.
Small birds can actually freeze to death in winter because they have insufficient calories to burn and keep their little bodies warm if they're stuffed with bread, and ducks and geese can develop angel wing from malnutrition that makes them incapable of flight.
Moldy bread should just be tossed - the mold is toxic for all animals. It's especially bad for small animals with rapid metabolisms.
Sorry to be a downer about this.

(Goodness knows my stupid parrot loves bread, but he only gets it when he steals bits of my burgers from me, and only if he takes full bites to include the meat and condiments. He loves him a good McChicken. Plus he gets that quite rarely - and will bypass bread for french fries any day of the week.)