It is implied (at the end of the story) that Bruno's parents eventually guessed how their son's demise came about. From the pile of clothes and boots Bruno left by the fence, Bruno's father could only come to one conclusion: his son, Bruno, had died in the gas chambers.  

In Chapter 20, we learn that it was a soldier who discovered the pile of clothes and boots by the fence. The soldier had alerted his Commandant to the discovery immediately. However, a year passed before Bruno's father figured out how Bruno died. After going back to the spot he was shown a year before, Bruno's father found that the base of the fence in that spot was not attached properly to the ground. The anomaly left a small gap large enough for a very small boy to slip through to the other side.

As his eyes followed the possible trajectory of Bruno's movements, Bruno's father came to the stunning conclusion that his own son had fallen victim to the system of oppression he championed. This realization so devastated Bruno's father that he found little appetite to continue his previous cruel campaign. Thus, when soldiers came for Bruno's father, he willingly went with them, "happy to do so because he didn't really mind what they did to him any more."