Some Twitter users blasted Bennett for posting the video, saying it was "weird" and "creepy".
Others suspected Bennett, the deputy Prime Minister, would be reprimanded tomorrow when she returned to work.
"You sure he wanted you to share that?" one user asked Bennett.
"High altitude makes you feel like throwing caution to the wind," another commented.
"Bless his little dad dance," said a third.
Bennett earlier posted a photograph of her and English with former Prime Minister Sir John Key at Eden Park.
Key was pictured holding a glass of beer.
"The one with the beer is not running in this year's election," Bennett wrote.
English has been the subject of many jokes on social media since he became Prime Minister.
In April he posted a photograph of himself making dinner for his family on Facebook.
The dinner was homemade pizza.
"Cooked dinner for the family last night - like if you agree with tinned spaghetti on pizza!" he wrote.
The mocking of English on that occasion went global, with US late night comedian Jimmy Kimmel joking that the leader declared war on Italy with his canned spaghetti pizza.
"We can be pretty hard on Donald Trump but they have a guy in New Zealand, the Prime Minister of New Zealand, a gentleman named Bill English, who I think is even worse and I'm going to show you why," Kimmel said.
"Not only did he put (tinned spaghetti) on a pizza - he put it on a pineapple pizza. He put canned spaghetti and pineapples on a pizza.
"This mother... That is so offensive. That is an act of war.
"I think he just declared war on Italy ... and maybe Hawaii too."
Then in May English hit the headlines again after posting a video of his "walk run" exercise regime.
"People often ask what I do in my spare time, so I thought I'd show you first-hand," he wrote.
English then embarks on what he calls a "walk-run", explaining: "I walk up the hills and run the downhill bits."
