Does either imply a lesson, or a fantastical setting?
My understanding is that a fable involves (speaking) animals or other mythical creatures, while a parable does not. A moral is typical for both genres.
Wikipedia is more accurate in its wording:
Merriam-Webster basically agrees, but has a few points to add:
My observations - not an authority's formal definitions:
The most famous fables (Aesop's) all feature animal protagonists, but I don't believe that has ever been a requirement, Wikipedia notwithstanding. Any fictitious story with an implied or explicit moral, or lesson, and which is obviously contrived for the purpose of communicating that moral or lesson can be referred to as a fable. Example: http://www.userfocus.co.uk/fable/index.html
A parable is not necessarily contrived, but usually contains a relatively specific analogy. A fable is more vague and illustrates a more general principle or concept. The parables in the New Testament (per my poor memory) are all things that "could have actually happened like that".
Edit: Apparently the root of "parable" is from the Greek "to compare", so a specific analogy is definitely implied.
Fables are stories that feature animals, plants, or forces of nature that have been given human qualities.
They teach moral and ethical lessons, like how to behave or how to treat people.
Since the main characters are animals, they are a good way to introduce serious topics to children. Each animal represents a particular human fault or virtue, and what happens in the story is directly related to the animal’s personality.
Parables also teach moral and ethical lessons, but they only have human characters.
They are set in the real world, with realistic problems and results. They often have spiritual aspects.
So what would a story be that features a human interacting with a talking animal? A fable, since parables exclude unrealistic things like chatty foxes.