Sunaria Hamid-Howells, who suffered a "long sequence of abuse" in the weeks leading up to her 2012 death, would have become immediately unresponsive after being injured, Cardiff Coroner's Court heard.
On Wednesday assistant coroner Christopher Woolley said it was not possible to tell from the evidence whether her mother Katherine Howells or father Dilshad Hamid had inflicted the fatal injuries, but it had to be one of them.
Sunaria was rushed to hospital from her Pontypridd home on December 30, 2012, but died the following day from her injuries.
Mr Woolley said Sunaria was held onto "forcefully" by her clothing, which explained bruises and abrasions on her body.
"She would have become immediately unresponsive after the shaking and head trauma and this would have been obvious to whoever had done this," he said.
"She was subjected to violence at around 5.30pm that evening... (after) a long sequence of abuse towards Sunaria."
He found neither parent had told the full truth about her death or injuries.
An inquest was originally opened into Sunaria's death in October 2015 but was adjourned by Mr Woolley who said he heard evidence that a homicide may have taken place.
Prosecutors had since decided there was no realistic prospect of conviction and that nobody would face prosecution over the death, he said.
On Wednesday, Mr Hamid and Ms Howells denied they had harmed Sunaria, saying that they had loved their daughter.
But Sunaria's uncle and Ms Howells' brother, Michael Howells, called on one of them to admit what they had done.
He said: "The evidence suggests that either Katherine or Dilshad caused the injuries.
"One of you caused her death.
"Please, before we leave this court today, please one of you stick your hand up."
Mr Hamid told the inquest he was giving Sunaria a bottle at around 5pm on December 30 when her arms began shaking and she went limp.
Paramedics were called and Sunaria was rushed initially to the Royal Glamorgan Hospital but was then taken to University Hospital Wales, Cardiff, to the paediatric intensive care unit.
Outlining the evidence given by Ms Howells on the previous occasion, Mr Woolley said she had been cleaning the bathroom floor when she heard a "really disturbing cry".
Questioning her, Mr Hamid asked: "What happened to my baby?"
In a heated exchange, Ms Howells said Sunaria was "absolutely fine" when she had put her down in her basket.
She said: "I would not hurt my baby - I loved that little girl."
The inquest heard Sunaria had been taken to hospital with a rash on two occasions and had suffered a fractured leg at some point before Christmas.
A friend of Ms Howells also found a lump on her head on Boxing Day, while family members noticed bruising on her in the weeks beforehand.
Mr Woolley found the bruises, fractured leg and rashes to be non-accidental.
The inquest heard Ms Howells told police Sunaria had been dropped during a trip to Bluestone Resort in Pembrokeshire in November, had fallen from the sofa twice and had fallen from her rocker on Christmas Day on to carpet and had a "soft landing".
Mr Woolley said doctors told the inquest that none of these incidents could explain the severity of the injuries found when she was admitted to hospital on December 30.
There was also evidence of an old subdural haematoma aswell as the one that caused her death.
Mr Woolley asked if Mr Hamid had ever harmed, shaken or dropped Sunaria or twisted her limbs.
Mr Hamid said "of course not" and added that he had not seen anyone else hurt her.
At the end of his evidence he added: "I want to know what has happened. I have never done anything wrong with my baby except loving her."
Mr Woolley said: "One person did the unlawful act. It is not possible on the evidence to determine whether this person was Ms Howells or Mr Hamid but I find that it could only have been one of them."
The inquest heard the medical cause of death was brain swelling and subdural haemorrhage.
Mr Woolley added: "I'd like to offer my condolences to all who genuinely cared for Sunaria during her short life."
Speaking after the hearing, Mr Hamid, 28, said he wanted to know what had happened to Sunaria and said it "was not fair on the baby".
He said of his daughter: "She was happy, she was beautiful. I have never seen in my life how I was happy when I was having her.
"It just goes away for nothing. Until I die I'll be thinking about what is going on."
Mr Hamid said he had requested an interpreter for the hearing but had not been allowed one and added that the legal process had caused problems for him and his new partner, Ayesha Khan, as they had not been able to get married aside from in an Islamic ceremony and feared any children they had might be removed from them.
Ms Howells declined to comment when approached outside the court.
A spokeswoman for South Wales Police said the force had "fully investigated the death of Sunaria and presented all the evidence to the CPS and the coroner".
She added: "Our thoughts remain with the many members of Sunaria's family who loved her and continue to mourn her loss."