A "parable" is a story that has a specific message in mind.
A parable is a short tale that illustrates universal truth...
Some parables are religious, but many are not. A parable might be an extended metaphor. Except for a magical aspect, fairytales are often parables also. For example, in "Bluebeard," the new wife goes into the forbidden room only to find the bodies of her husband's previous wives. Although she is saved in the end, the message is that "curiosity killed the cat."
Aesop's Fables are also parables, including stories like "The Fox and the Crow," and "The Grasshopper and the Ant." George Fyler Townsend, who wrote a preface to a translation of one version of the fables, described the parable as
...the designed use of language purposely intended to convey a hidden and secret meaning other than that contained in the words themselves, and which may or may not bear a special reference to the hearer or reader.
Many well-known parables were told by Christ, and are recorded in the Bible. They were about common people, events or things. The parable of the Good Samaritan in told in the Gospel of Luke. It is about a stranger (the Samaritan) who cares for a mugged man by the roadside, even though his people and the Samaritans do not get along. Still the passerby offers assistance. The message, which is non-sectarian (not reflecting a particular religious ideology), offers positive advice or an example. In a biblical context, this story deals with the religious theme: love your neighbor. As a parable standing alone—used as a cliché—it describes anyone who helps a stranger out of kindness.
A morality play was a dramatic presentation that was produced by the Roman Catholic Church in medieval times. It was allegorical in nature. This means it may have had a literal message, but it would also have had a deeper, symbolic meaning. These early dramas were popular from the 14th through the 16th Centuries. Morality plays weren't strictly based on the Bible; the concentration was on morality as opposed to spirituality. Everyman is not the earliest version of a morality play, but it is easily one of the most well-known.
Both forms are intended to share a concept with regard to living a righteous life. The morality play...
...[grew] out of the religiously based mystery plays of the Middle Ages...
However, while some parables are religious in nature, this is not necessarily the case for all parables. Aesop's Fables is one example of a collection of stories that offer advice and a "moral" without being religiously inspired.
George Fyler Townsend, translator's preface to Aesop's Fables, Belford, Clarke & Co., 1887.