In a video statement, Ed and Paula Kassig said they were proud of their son and the aid work he had done.
The statement follows the release of a video by the militant group on October 3, which showed the killing of British aid worker Alan Henning.
That video ended with a threat to kill 26-year-old Peter Kassig.
It was the fourth such video released by the ISIS group.
Previous victims were American reporter James Foley, American-Israeli journalist Steven Sotloff and British aid worker David Haines.
Peter Kassig’s parents said he had been working for the relief organization he founded, Special Emergency Response and Assistance (SERA), when he was captured a year ago on his way to Deir Ezzour in eastern Syria.
In their appeal, they said: “As Muslims around the world, including our son Abdul-Rahman Kassig, celebrate Eid al-Adha, the faith and sacrifice of Ibrahim, and the mercy of Allah, we appeal to those holding our son to show the same mercy and set him free.”
It is believed Peter Kassig changed his given name to Abdul-Rahman and converted to Islam while in captivity.
The family has heard from former hostages that his faith has provided him comfort.
ISIS militants may hold many more hostages.
On October 3, the father of John Cantlie, a British photojournalist held by the group, appealed for his release in a video, describing his son as a friend of Syria.
The ISIS group has its roots in al-Qaeda’s Iraqi affiliate but was expelled over its brutal tactics and refusal to obey orders to confine its activities to Iraq.
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