An animal story with a moral

One word substitute is Fable.

Fable: a short story, typically with animals as characters, conveying a moral.
Tale: a fictitious or true narrative or story, especially one that is imaginatively recounted.
Anecdote: a short amusing or interesting story about a real incident or person.
Parable: a simple story used to illustrate a moral or spiritual lesson, as told by Jesus in the Gospels.