What do the bridge, cats, goats, and pigeon symbolize?

The bridge could be said to symbolize a means of escape for the old man from his present life, a chance to start a new life in another town. But it could just as easily symbolize an uncertain future. The solider narrator tells the old man that the road at the end of the bridge leads towards Barcelona. But as the old man doesn't know anyone there, he stays put. He's already been forced to leave his hometown due to the ravages of war, and feels that he's come to the end of his life's journey. The bridge may represent the chance of a new beginning, but not to an old man who's pretty much given up on life. In that sense, this is one bridge too many, an insurmountable hurdle that the old man cannot and will not overcome.
The old man's extensive menagerie—his cats, his goats, his pigeons—could be said to symbolize hope. As long as the old man was able to take care of his animals, he had some hope in life, something to live for. But once he was forced to leave his animals behind in San Carlos, the old man had nothing more to live for, and what little hope he had effectively died inside him.