A fable is a short fictional story that usually features animals, plants, or other non-human characters given human traits, and that teaches a practical moral—often stated explicitly at the end. Classic examples include Aesop’s “The Tortoise and the Hare.”
A parable is also a brief teaching story, but it typically uses human characters and familiar situations to convey a moral, ethical, or spiritual lesson rather than anthropomorphized animals or objects.
In short: fables personify animals or nature to deliver a moral; parables keep the cast human and often aim at ethical or religious insight. Source: Fable — Wikipedia.