A fable is a short story with a moral that has animals as the main characters. A parable is a short moral story without animals as characters. Fables and morals have been around for centuries, but modern moral stories, such as "Fables of Our Time" by James Thurber, also exist.
All are available in print or online or both. Parables were among the first to be published with the printing of the Bible. Some of the most ancient fables are those of the "Panchatantra" of India, which gave rise to the Arabic tales of "Kalila and Dimna." "Aesop's Fables" and the fables of Jean de la Fontaine are better known in the Western world. Online, some examples of fables and modern moral stories can be found on the following websites; aesopfables.com, taleswithmorals.com. Some digital fables are available on such websites as https://archive.org/details/kalilaanddimnao00almgoog.