To answer this question, it is first necessary to establish what an antagonist is. In any given story, the antagonist is the harbinger of trouble and the cause of the disruption to the protagonist's life.
In the case of "Once Upon a Time," one might argue that the antagonist is not a person, but an emotion: fear. It is thanks to fear that the barbed wire with which the couple's son will be mangled is installed. In a tragically ironic sense, it is their fear and wish to protect their family that results in their son's injury.
The main characteristic of an antagonist is that he or she causes harm. While it is unusual for feelings or emotions to be an antagonist, it is the couple's fear that directly leads to their son's horrible accident. An antagonist is also said to be the provider of an obstacle in the protagonist's life. In this case, the family's fear leads to the construction of a physical obstacle, which proves to be their downfall.
While the barbed wire and wall were designed to protect the family by preventing intruders from entering their property, the son climbs into the barbed wire and is himself injured. In this case, the fear that led to the wall's construction inadvertently causes the son's tragic accident.
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