Let us remember that symbols can take the form of objects, actions or characters, and in this excellent fairy story I would argue that the four main symbols are the old man, his wife, and the two very different treasure chests they receive at the end of the story.
The old man seems to symbolise generosity and goodness. Note the way he looks after the sparrow and cares for her, protecting her from predators and giving her shelter. The old woman, on the other hand, clearly is very different from her husband. Note how the text describes her:
...the wife being greedy and quarrelsome when anyone came her way that she could possibly quarrel with.
The old woman therefore seems to symbolise greed. Note the didactic function of this tale. The two treasure chests at the end seem to symbolise the just desserts that you can expect to receive if you commit your life to generosity and goodness, as in the case of the old man, or to greed and evil, as in the case of his wife. The symbolism of the wife's treasure chest is of course the most explicit:
At length, however, the lock gave way, and the lid flew open, when, O horror! instead of gold and jewels, she saw before her serpents with glittering eyes and forky tongues. And they twined themselves about her and darted poison into her veins, and she died, and no man regretted her.
The venomous snakes that are in the chest are a symbol of the way that greed can ensnare us and bring us to our own ruin. The wealth that the old man receives, on the other hand, is a symbol of what generosity towards others will win us.