Cautionary tales
Fabular tales
Moral and philosophical tales
Morality tales
Philosophical and moral tales

Parables.

Wheeler, K. Parables and fables : from symbolism to allegory?, via WWW, Apr. 4, 2013 (A fable is also a brief story illustrating a moral. Unlike the parables, fables often include talking animals or animated objects as the principal characters. The interaction of these animals or inanimate things reveals general truths about human nature, i.e., a person can learn practical lessons from the fictional antics in a fable. However, the lesson learned is not allegorical. Each animal is not necessarily a symbol for something else. Instead, the reader learns the lesson as an exemplum--an example of what one should or should not do.)
Merriam Webster's encyclopedia of literature, 1995 (Fable. A narration intended to enforce a useful truth; especially, one in which animals or inanimate objects speak and act like human beings. The Fable differs from the ordinary folktale in that it has a moral that is woven into the story and often explicitly formulated at the end.)
Quinn, E. A dictionary of literary and thematic terms, c1999 (Fable: A short narrative in prose or verse in which the action of the characters, usually animals, conveys a moral lesson.)
Baldick, C. The Oxford dictionary of literary terms, 2008 (Fable: A brief tale in verse or prose that conveys a moral lesson, usually by giving human speech and manners to animals and inanimate things. Fables often conclude with a moral, delivered on the form of an epigram. A very old form of story related to folklore and proverbs, the fable in Europe descends from tales attributed to Aesop)
LCSH, Oct. 22, 2014 (Fables. BT Exempla; Fiction)
GSAFD, 2000 (Fables. UF Cautionary tales and verse, Moral and philosophical stories, Morality tales.)
Looking with Robert Gardner, 2016: p. 204 (fabular tales of high school life)
Historicizing emotions, 2018: p. 299 (fabular tales of sacrifice and loss among friends)
Merriam-Webster dictionary online, Nov. 14, 2018 (fabular: of, relating to, or having the form of a fable; fable: a fictitious narrative or statement: such as a) a legendary story of supernatural happenings b) a narration intended to enforce a useful truth, especially : one in which animals speak and act like human beings c) falsehood, lie)

Stories intended to teach moral lessons, and whose main characters are generally animals or inanimate objects that speak and act like human beings. For short, simple stories that convey a moral lesson, and whose main characters are generally humans, see