A few words of caution in writing a symbolic story are to be careful to avoid cliches and symbols that are so overt that they are almost unnecessary. Another tip I have (for any writer) is to immerse yourself in literature that will enhance/inspire/coach your writing.
In addition to the classic authors mentioned in previous posts, you might also want to consider some slightly more modern authors who have been successful with symbolic stories: CS Lewis and the Chronicles of Narnia; JK Rowling and the Harry Potter series, or even Stephanie Meyer and the Twilight series. Despite what your personal feelings on these authors and their works may be, they have all been very successful and symbolism is prominent in their works as well.
Since American literature is replete with symbolism, look to the masters. As previously mentioned, Nathaniel Hawthorne is the initiator of symbolism in American literature. His The Scarlet Letter is a narrative that revolves around the various interpretations of this symbol. Other greats include Ernest Hemingway, who uses symbolism to give greater meaning to his minimalist prose.
What is wonderful about symbolism is the fact that there can be more than one meaning; thus, the reader is afforded different levels of meaning in a narrative that contains symbols, making richer the narrative. (Consider reading Hemingway's "Hills Like White Elephants" to see how the hills that appear white have two or three symbolic meanings. Symbols can be outside the immediate narrative, reflecting the meaning of dialogues, as in Hemingway's story.)
Writing a symbolic story depends upon building the character development; the conflict and resolution; the setting; and/or the theme(s) of a story around one or more symbols. Hawthorne is noted for the exceedingly detailed symbolism he works into his novels and stories like The House of the Seven Gables, "The Minister's Black Veil," and that chilling tale, "Young Goodman Brown." Let me give you a hypothetical instance of uses symbols in writing a symbolic story--or--writing a story developed around symbolism.
Let's consider character development. We'll say your female protagonist had a rough early life in which she was under-appreciated and made to fill the place of mother to her three older brothers while her mother and father stepped to the background and sought their own happiness (with plausible excuses of course). Your heroine will have a good heart but it will be crusted over with hard barnacles that prevent the true milk of human mercy and compassion from flowing--she'll be rather too harshly bossy and manipulative.
If you give her a symbol that connects her to, say, the Cascade Mountains or the Rocky Mountains, perhaps through a cherished pendant that she always wears or a collection of mountain posters and pictures that she cherishes, you've automatically created a dimension to her character that indicates she will struggle up a long climb to the liberation of her true humanity but that by the end of the story, she will be breathing the pure, rich air of the pinnacle of the human spirit on the mountain top.
This is how to write a symbolic story. You choose the element of the story in which you want to weave the symbol(s). You pick a suitable symbol for the point you want to make; the image you want to evoke; the emotion you want to arouse; the resolution you want to attain; etc. You weave it into the circumstances of your story in a life-like way. Then, when your writing is through, you will have written a symbolic story.