When you pull chestnuts out of the fire, you end up doing someone else’s dirty work. The other person is in such a tight spot that you feel compelled to help him, often putting yourself in danger. Another expression that has more or less the same meaning is ‘cat’s paw’.
As expected, Druv’s rich parents pulled chestnuts out of the fire for him once again.
The expression has its origins in a fable. In the story, a cat and a monkey are sitting around a fire watching their master roast chestnuts. The monkey wants to eat the nuts, but he doesn’t want to pull them out of the fire because doing so would burn his hands. While the master is away, the cunning monkey convinces the poor cat to do his dirty work for him. The cat scoops out the nuts one by one and in the process burns his paw. The selfish monkey eats all the nuts leaving nothing for the silly cat.