Both Epimetheus and the father are the two characters in the story that try to avoid bad situations., because Epimetheus tells Pandora to never touch the box, and the father wishes his son to disappear.
Both Pandora and the White family were warned to not toch the objects that were bad. The Sargent-Major warned the White family, and Epiimetheus warned Pandora.
In "Pandora's Box" al the characters are gods or goddesses, compared to "The Monkeys Paw" where no one in that story is a god or goddess.
In both of the stories the consequences were caused by different things, in "Pandora's Box" wishes didn't cause evil to go into the world, and in "The Monkey's Claw" Opening a box didn't cause their son to die or to come back from the dead.