In June this year, Dickason was sentenced to 18 years in prison for killing the girls at their Timaru home.
She smothered 6-year-old Lianè and 2-year-old twins Karla and Maya in their beds on September 16 in 2021, three weeks after the family had arrived in New Zealand from South Africa.
The 43-year-old had pleaded not guilty to the murder charges, her legal team mounting a defence based on insanity and infanticide.
But on August 16 last year and after more than 14 hours of deliberations, a jury found Lauren had murdered them.
Justice Mander imposed no minimum non-parole period, saying her severe mental illness was not only a contributing factor to her actions, but the cause.
Dickason will be eligible for parole after she has served a third of her sentence.
She was initially sentenced to an in-patient care and treatment unit as a special patient because of her mental condition, in accordance with the Mental Health Compulsory Assessment and Treatment Act.
No date has been set for the appeal.
Dickason's lawyer has been approached for comment.
