Triple axe murderer Henri van Breda's application for leave to appeal his conviction and sentence was denied in the Western Cape High Court on Monday.
Judge Siraj Desai ruled against the convicted killer, saying the defence had not provided compelling reasons for the application to be allowed.
He said defence advocate Pieter Botha's arguments were similar to those made during the trial, and that the evidence and facts supported the conclusion that Van Breda was behind the 2015 attack.
Botha told News24 after the ruling that they will now be petitioning the Supreme Court of Appeal.
Van Breda's application was heard last Tuesday.
Botha had argued that his client - a "very young person" - was facing life in jail based only on circumstantial evidence, maintaining there was a realistic possibility that unknown assailants could have gained access to the De Zalze Estate on the night of the gruesome murders.
Van Breda alleged that an intruder, wearing a balaclava, gloves and dark clothing, was behind the murders, and that that he had heard other voices of people speaking Afrikaans in the family's Stellenbosch home in January 2015.
He claimed that, after a fight with the axe-wielding assailant who was also armed with a knife, the man had escaped.
The defence submitted that there was a realistic possibility, and not "merely an unlikely possibility" as found by the trial court, that unknown intruders could have gained access to De Zalze Estate that fateful night.
Botha believed that, should an appeal court uphold the conviction, there was a reasonable prospect that it would find that the prosecution had failed to prove planning or premeditation.
The damning evidence of experts who believed Van Breda's injuries had been self-inflicted had also not been proven beyond a reasonable doubt, he said. Botha also contended that the court had erred in rejecting his client's version of how the superficial cuts to his arm and chest were inflicted.
'Like pieces of a puzzle'
During the application, Desai during remarked that the evidence in this case "fit like a mosaic, like pieces of a puzzle".
He said the sentence would have been different if Van Breda had shown that the murders were a product of a disturbed mind.
Van Breda, however, had maintained his innocence.
Botha said no light had been shed on the motive for the attacks, especially given that the Van Bredas were a "happy family". But Desai had countered that the same could be asked about the motive of an unknown intruder.
Prosecutor Susan Galloway agreed with Desai's finding that the attacks were premeditated, as Van Breda would have had to had gone downstairs to fetch the weapon before returning to the upper floor.
The axe had been aimed at his victims' heads, which showed direct intent to kill, she had asserted.
Desai in May handed Van Breda three life sentences for the murders of his parents and brother, 15 years for the attempted murder of his sister and one year for obstructing the course of justice.
Van Breda, who is serving his sentence in Drakenstein Prison in Paarl, was not present in court on Monday, and during the application.