The new footage of Alon Ohel was released Monday by Hamas, shortly before Rosh Hashanah - the Jewish new year.
Alon's parents, Idit and Kobi Ohel, said their family was "shaking and in pain" after seeing the video. "It is clear that Alon is losing sight in his right eye and he looks thin and frightened. We demand that as a condition for any negotiations or further assistance to Hamas, eye doctors examine Alon and provide him with treatment.
"On the eve of Rosh Hashanah, in these days of repentance, the fate of young Jewish citizens of Israel now lies in the hands of the prime minister and the cabinet. Do not break the heart of the people of Israel. The most important national news today is the return of Alon and all the hostages to their homes," they said.
The parents added that the contents of the video amounted to psychological terror and urged the public not to share it.
Ohel appeared alongside Guy Dalal in a video released earlier this month. His family confirmed its publication at the time, after reviewing it.
