How did his parents know Bruno was in the gas chamber?

In chapter 19, Bruno agrees to help his friend Shmuel find his father in the concentration camp. In order for Bruno to enter the concentration camp without attracting attention from the Nazi guards, Bruno takes off his clothes and boots before putting on the "striped pajamas" and climbing underneath the fence. After Bruno and Shmuel tragically die together in the gas chambers, Bruno's father has his soldiers search the camp and its premises for signs of his son. In chapter 20, the soldiers end up discovering Bruno's pile of clothes and his pair of boots on the outside of the fence.
Time passes after the soldiers discover Bruno's clothes near the fence, and Bruno's mother eventually moves back to Berlin. One year later, Bruno's father forms a theory and returns to the location where Bruno's clothes were found. He then notices that the bottom of the fence is not properly attached to the ground and that there is a space large enough for a small person to crawl through. Bruno's father then contemplates deeply the possibility that his son died in the gas chambers and immediately loses his balance.

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."