The concept of allegory you are talking about is called personification and has been mainly used in literature in medieval times and the baroque.
There are other types of allegories, like in the Hebrew Bible, let's say Psalm 80, talking about the vine that stands for Israel. Things, as the vine, and actions can be allegorical.
In arts there are many allegorical pictures. A personification would be Justice as a blindfolded woman with scales.
Fables are a subcategory of allegories, as are parables. Both probably are characterized by shortness.
All three categories are forms of writing, art, or spoken utterance that encourage readers to look for meanings beyond what is said.
As for the difference between fable and parable: the fable, as the OP says, has animals, plants, or objects acting. It therefore has to anthropomorphize, while a parable draws its images from human interaction mostly.
Therefore a fable most the time is more schematic in build and easier to decipher. Parables often allow for different ways of deciphering. Looking at Kafka, Brecht, or biblical parables, it is clear that there is often a key, hint, or explanation needed to decipher the parable.
This might be because a fable describes something that is naturally not possible, as the actors are anthropomorphized, creating its moral effect using striking simplification, while the parable describes a naturally possible incident, allowing for more complexity due to acquaintance, creating its moral or parabolic effect through surprise. Fable and parable therefore have much in common and overlap greatly.