In the spoof, the female Korea expert calmly unpacks the political situation in South Korea after the country's president was impeached last week, all while she feeds her daughter, makes a chicken dinner, and scrubs a toilet.
Later, she even defuses a bomb after a swat team rushes in and hands her a pair of wire cutters.
Many of the show's fans took to Facebook to praise the spoof.
"It's so funny... my first thought when that father shoved his child away was that a lot of moms would have totally been able to handle the kiddos while keeping her stroll on! Lol!" one fan wrote.
But some viewers interpreted the sketch as sexist.
"Are you assuming the dad is some sort of non-cooking, non-cleaning stereotype just because he forgot to lock the door during a serious tv interview?" one viewer asked.
The original video was viewed by millions since it was uploaded by the BBC last Friday.
Kelly, who is from Cleveland Ohio, focused entirely on the camera as he attempted to blindly hand off his daughter, who swaggered into the room during his interview, curious to see who he was talking to.
His mortification soon doubled as baby James excitedly made his way into the room under his own power in a walker.
The children's mother, Jung-a Kim, then came skidding through the door to scoop up the kids.
Eventually, she managed to get them both out, and the interview continued.
When the interview finished, BBC broadcaster James Menendez told Kelly: "There's a first time for everything. I think you've got some children who need you!"
Kelly admitted on Wednesday he was mortified at the time but in hindsight he could see the funny side.
He said his feelings about the incident had gone from "surprise and embarrassment" to "amusement" and finally "love and affection".
"It was terribly cute. I saw the video like everybody else and it's really funny," Kelly told the Wall Street Journal.
He said the reaction on social media had been astonishing - and mostly positive - and he had been forced to switch off Twitter and Facebook alerts and put his phone on airplane mode.
"I'm not even going near YouTube or Reddit or whatever those other sites are," said Kelly.
His wife said: "It happens all the time but not like this. This was the first time it happened during an interview."
In the immediate aftermath of the video going viral many people on social media assumed she was his nanny or maid, rather than his wife.
Some people accused those who assumed she was the nanny of being racist, leading to some fairly aggressive Twitter fights.
In an interview with the BBC Prof Kelly said: "We were pretty uncomfortable with it."
But Jung-a played it down and said: "People should just enjoy it, not argue over these things. I hope they stop arguing."
