A Poison Tree is not a biblical story, although a tree is often mentioned in Genesis as the harborer of knowledge and the place in which sin lurks. This premise colors the poem but the it is not found literarily in the Bible. However, "The Poison Tree" can definitely be classified as a parable.
A parable is a story told with the purpose of teaching a moral, or a lesson. In this case, the Poison Tree is allegorical to growing hatred within our hearts, and letting its roots anchor within our souls hence producing nothing but poisonous feelings, and turning us into offensive beings.
In this parable, a man is harboring a resentment that he feels against an antagonist. He does not allow himself to analyze the feelings, nor does confront them. He simply allows them to fester within, and grow bigger and bigger. As they grow, the main character becomes more and more miserable, until he finally sees his enemy dead, poisoned by his own anger.
Therefore, the purpose of this parable is to show that harboring anger, rancor, and holding grudges leads us nowhere. In fact, cutting the bad emotions from their root (as if a tree) will prevent us from harvesting the products of such negativity.