Timothy, not his real name, had spent most of his short life living with his grandparents in a crowded but loving family home in Sydney's south.

Relatives say he was happy and healthy, rough-housing with his cousins outside, playing video games and chatting on the telephone.

But on August 6, 2014 – just seven weeks after his mother and stepfather brought him to their Oberon home – he was dead.

Timothy's mother and stepfather are on trial for murder in the NSW Supreme Court.

On Monday, the court heard that the pair had made admissions to inflicting horrific abuse on the boy during the short time he was under their care.

"I had so many thoughts about killing Timothy because I loved him but I just didn't connect with him," the mother told police.

"Don't get me wrong, I did love that boy … but there was a part of me that hated him because he looked like his father, because he is his father's child."

The boy was beaten repeatedly with a wooden spoon and a ball was taped to his mouth and his hands bound behind his back.

He was put into a portable cooler filled with ice and the lid pressed down on top of him as he screamed and desperately tried to get out.

In the days before Timothy died, his mother admitted she had rammed a cupboard door into his head because she was "so pissed off" and "just wanted to get some sleep".

"He was standing up against the wall and I just kept poking his forehead because he wouldn't stand up straight and his head kept banging on the wall," his mother told police of another incident that happened days before he died.

"I don't know what I did to him because I was really pissed off but he ended up in like a catatonic state."

But the pair say they are not responsible for Timothy's death.​ They say the injury that killed him occurred during a family visit to the park with their dogs.

"We were running around and then the dogs ran past him and knocked him over and then he went limp and he was still breathing and then started gurgling," the mother said in a recorded triple zero call.

Timothy died three days later in hospital. The cause of death was cardiac arrest and an autopsy detected multiple blunt force trauma injuries on his body, scabs on his face and serious injuries to his spine.

In her opening address to the jury on Monday, Crown prosecutor Margaret Cunneen said the pair's story was a lie.

Timothy, she said, was in no state to be taken to the park on the day he died and they had concocted a ruse to explain what had happened to him.

The mother's defence case is that the injuries Timothy suffered at her hands were not in any way related to the boy's death.

"Her case is, simply put, her son died as a result of what happened in the park, the injuries occasioned by a fall as a result of the rope tripping her son," her lawyer, Eric Wilson, said.

The court heard there are conflicting medical opinions about whether Timothy tripping, falling backwards and hitting his head could have caused the injuries that led to him going into cardiac arrest.

The stepfather's barrister, Alyssa Moen, is due to outline her case on Tuesday.