Trig, 6, who has Down Syndrome, is shown stepping on the back of the family's black Labrador while it lies on the kitchen floor in order to reach the sink. The dog appears unruffled.
The Facebook post by Palin, who was the Republican vice presidential nominee in 2008, drew more than 12,000 comments by midday Friday, ranging from cries of animal abuse to those seconding Palin's praise for the youngest of her five children.
"May 2015 see every stumbling block turned into a stepping stone on the path forward," Palin wrote on her Facebook page.
"Trig just reminded me. He, determined to help wash dishes with an oblivious mama not acknowledging his signs for 'up!', found me and a lazy dog blocking his way. He made his stepping stone," Palin wrote.
The message earned 50,000 "likes" from some of Palin's 4.5 million Facebook followers.
But some online commentators were infuriated.
"You want to run your state and the country, but you don't have the sense to tell your children to stop stepping on your dog's spine?" Kenneth Meador wrote.
"Yes the dog could have moved but the greater lesson is he's gonna get bit and hurt and the dog will get blamed instead of the child that is abusing the dog/ should a child ever be allowed to step on any animal ughhh," posted Amy Payzant Steward.
Other followers were full of praise for both Palin and Trig.
"Trig is a great problem solver ... and to all you "haters" out there spewing your venom ... this is a little boy with his dog who is doing nothing wrong! And Sarah I enjoy your quote for the new year!!" commented Linda Conley Eltzroth Darling.
The story of Romney having driven with the family's Irish Setter on top of his car, inside a dog crate, was discovered by the Boston Globe in 2007 and recounted repeatedly during Romney's failed 2012 Republican presidential bid. The dog incident occurred in 1983 when Romney was a rising star in the private equity world.
Palin, a former Governor of Alaska, was the running mate of Republican presidential nominee John McCain in 2008; McCain lost to President Barack Obama.