What would be an idea for a short story that parallels the themes of Animal Farm, but places it in a new and different context?
The most elemental approach for a short story that expands Orwell's ideas would be to continue the animals' allegory in a different setting. Perhaps, a sequel could be explored here. Could there be a new uprising in which Jones's role is taken by Napoleon and there is a new leader within the farm? A real interesting take would be to make an "Arab Spring" uprising in which one of the animals being controlled by the pigs and dogs ends up making some type of symbolic statement that compels the other animals to massive and uncontrolled rebellion. The "Arab Spring" condition is something that Orwell, himself, did not see. The individual and self- sacrificing actions of a fruit seller named Mohammed Bouazizi who had experienced enough sets himself on fire, and, in the process, he ignites the fires of change that toppled many a government.
Something similar to could be done in a short story based off of Animal Farm. One of the hens engage in an act of demonstrative and suicidal resistance. If one of the hens set themselves on fire, it is plausible that the entire farm would rebel and even Squealer would not be able to placate them. Placing the structure of Animal Farm in an "Arab Spring" context might be really interesting and timely. Continuing this could be showing the challenges of governance. This idea is being seen in Egypt, one of the Arab Spring nations. Showing the farm descending into chaos of leadership and examining the difficulty in establishing a stable political order could be another way in which the ideas of Orwell's work are applied in a different context.