A parable is a short true story that also teach a good moral lesson. It may involve animals or other inanimate objects taking on human features as in the case of the donkey and rocks in parables given by Jesus. A parable always involves truth even about the donkey and rocks for Jesus spoke about what God could do. We know God can do what he wants to do.
A tale is a short story about something or someone. A story can be true or untrue. Therefore, a tale can be true or untrue though people usually use the word to refer to an untruth.
A myth is something claimed to be true yet cannot be proven definitively to be true or untrue but is probably untrue.
A fable is a short story which involve animals or other inanimate objects which speak or take on other human, or animal, or other untrue features.
The purpose of a fable is to teach moral lessons. Hopefully it teaches good morals. Yet, there are fables that teach bad morals.
If the fable is presented as for entertainment purposes without any claim of being true then it is a righteous fable. An example of righteous fable is an episode of the cartoon Charlie Brown in which only good morals are presented. Charlie Brown involves a dog named Snoopy who has human thoughts. Snoopy does not talk to other humans but his human-thoughts are presented in a thought baloon for viewers to see. There is no claim that dogs truly can talk.
If the fable is claimed to be true and presented as such then it is a wicked fable.
A fairy tale is a fable that assigns characteristics such as magical and/or unusual power, beauty, perfection and other desirable but humanly or otherwise impossible characteristics to a character called fairies. Fairies do not exist.
The bible speaks of fables in 1 Tim 1:4; 4:7; 2 Tim 4:4; Titus 1:14; and 2 Peter 1:16.
The Greek word translated fables in these scriptures (KJV) is mythos (Strong’s G3454). Hence, these scriptures speak of unrighteous fables equated with myths.
Click here for info on a kind of fairy tale fable called Santa Claus.