The footage from the Taliban-aligned Haqqani network shows Canadian Joshua Boyle and American Caitlan Coleman with two young boys, who appeared healthy.
In the clip Ms Coleman begs for an end to their "Kafkaesque nightmare".
The couple were abducted in Afghanistan in 2012 during a camping trip, while she was pregnant.
The insurgent group is demanding the release of three of its prisoners in Afghanistan.
In the four-minute video, the couple were shown with two young boys sitting on their laps.
Ms Coleman, 31, said: "We have waited since 2012 for somebody to understand our problems."
She appealed to President Barack Obama and President-elect Donald Trump to rescue her family.
"My children have seen their mother defiled," she added, without clarification.
In the footage, uploaded to YouTube on Monday, Ms Coleman also made an unexplained reference to her two "surviving children".
The couple last appeared in a video in August, without the children.
Michael O'Shaughnessy, spokesman for the Canadian Department of Global Affairs, said officials were "deeply concerned" for the couple and called for their unconditional release.
Taliban spokesperson Zabullah Mujahid said: "We are currently investigating about the video - at this stage we don't confirm or deny the release of the video."
There was also no immediate reaction from the US State Department or the Afghan government.
Mr Boyle and Ms Coleman set off in July 2012 on a hiking trip that took them through Russia, Central Asia and into war-torn Afghanistan.
An Afghan insurgent group has released video of a kidnapped North American couple, featuring two children apparently born to them in captivity.