The single mother's 3-year-old son is featured in the preview for the Lifetime programme calling his aunt Willow a "f*ggot."
Palin and her sister react by gasping and then giggling at the child's outburst, with Palin admitting that she is "doing a terrible job disciplining Tripp… I know he's going to continue to push the boundaries."
The 21-year-old and Lifetime were recently hit by a lawsuit for defamation of character and breach of privacy by Stephen Hanks, a man she had a verbal fight with in a bar while filming her show.
She allegedly accused the man of being homosexual in a derogatory manner when he openly criticised her mother, politician Sarah Palin.
Hanks, who is seeking over $75,000 (£48,206) in damages, also alleges that Palin said of his male companion: "That's a nice wife you have there."
He has also launched a second lawsuit against Palin, Lifetime's parent company A&E and Life's a Tripp producer David McKenzie claiming unspecified punitive damages for emotional distress.
Lifetime has pushed Palin's show from primetime to late night after less than stellar ratings.