The key feature of a parable is that it uses a situation, which feels very simple to make a more complex or general point, often a moral one. (Also, unlike a fable, a parable does this with people, not animals.) This classification defines "The Gift of the Magi," which is a remarkably simple story. It boils down to a few bare essentials: Della and Jim are poor, but love each other very much; they each want to buy the perfect Christmas gift for each other; they each have one prized possession which they give up to buy the other a present, and the presents they buy are meant for the prized possessions they've sacrificed. You don't need to know almost anything else about the story to "get it," and there's very little in the story itself that doesn't serve to develop one of those elements.
That there is actually something specific to get is the other reason "The Gift of the Magi" is a parable: it has a point, and yes, it is a moral one. This story is about what it means to give a gift. All of the elements of the story serve to bring that point across. And yes, the slightly "preachy" tone of the story is part of the parable. That last paragraph especially, which is just a slightly more stylish version of the "moral" that predictably comes at the end of an Aesop fable.