They are battling to keep their terminally ill 11-month-old son on life support so they can take him overseas for the treatment, a step opposed by his doctors on the grounds he may suffer without experiencing any benefit.
The judge, Justice Nicholas Francis, said he was keeping an open mind over his previous ruling that further treatment would be futile. "If there is important new evidence that suggests my decision should be changed then I will change it," he said.
But he insisted the new evidence must be significant for him to reopen the case.
About two hours into the hearing, Charlie's parents abruptly left the court after disagreeing with a comment by the judge.
Yates interrupted Francis as he said that the parents had said they would not want to prolong Charlie's life in its present state, only if there was hope of improvement. "I never said that!" she exclaimed from her seat behind her barrister.
The judge attempted to clarify that one or other of the parents had said it, but both rose and left the court. They returned to the courtroom after the lunch break.
Under examination by the barrister for Charlie's parents, a doctor testifying by video-link from the United States -- who cannot be identified by court order -- supported some key points of the parents' case, saying that the baby's MRI scan did not necessarily indicate structural damage to the brain.
The expert estimated there was an "11 to 56% chance of clinically meaningful improvement" in muscular function with the proposed treatment. The doctor also asserted that keeping Charlie on ventilator would not cause him harm because he did not seem to be in any significant pain.
Charlie's mother responded to the testimony with a thumbs up.
Hospital: 'No quality of life'
Charlie, who was born apparently healthy in August, has been treated at Great Ormond Street Hospital (GOSH) in London since October, when he began suffering the effects of mitochondrial DNA depletion syndrome
, a progressive disease that causes muscle weakness and loss of motor skills.
In its position statement to the court Thursday, the children's hospital said it had requested the new hearing last week
because the parents said they had new evidence that might change the judge's findings. But its own view has not changed.
It described Charlie as "a beautiful, tiny baby afflicted by one of the cruelest of diseases," which has left him "with no muscle function at all now and deprived of his senses, unable to breath and, so far as can be discerned after many months of encephalopathy, without any awareness."
GOSH paid tribute to the "great fortitude and devotion" of Charlie's parents. But, it said, Gard and Yates "fundamentally believe that they alone have the right to decide what treatment Charlie has and does not have," whereas the hospital "believes in its core that every patient is his or her own, unique and special person and that it owes a duty of care to each."
The hospital said it had treated more than 1,000 patients with mitochondrial disease and offered nucleoside treatment where appropriate. But, unlike his parents, it does not believe that Charlie can be helped.
"It has been and remains the unanimous view of all of those caring for Charlie at Great Ormond Street that withdrawal of ventilation and palliative care are all that the hospital can offer him consistent with his welfare," it said.
"That is because in the view of his treating team and all those from whom GOSH obtained second opinions, he has no quality of life and no real prospect of any quality of life."
Cross-examined about the potential treatment by the barrister for GOSH, the US-based doctor said there was "a small but significant chance of improvement in the brain." When asked by the judge, the doctor expressed willingness to come to Britain to see Charlie.
The judge briefly adjourned to let the barristers work out whether the US-based expert could fly to the UK to visit Charlie, but GOSH's lawyer issued objections to the plan.
Francis then set a hearing for Friday afternoon. The judge said no evidence would be presented and not to expect his decision at that time. The court session will be procedural, aimed at establishing the the parameters for a meeting among the experts.
Earlier in the hearing, the judge appeared angry that there had been no independent measurement of Charlie's head circumference since Monday, saying he wanted that resolved within 24 hours. This matters because whether the baby's brain is growing is a factor in the case.
Meanwhile, the family's barrister, Grant Armstrong, responded vigorously to a remark by the judge that the proposed treatment has never been tried on mice with Charlie's condition, let alone humans.
"As I understand it, there are only four people in the world with this condition -- you can't do a controlled trial," he said.
'We are still fighting'
After Yates and Gard first entered the court Thursday, family friend Alasdair Seton-Marsden read a statement on their behalf.
"We are continuing to spend every moment working around the clock to save our dear baby Charlie. We've been requesting this specialized treatment since November, and never asked the hospital, the courts, or anyone for anything except for the permission to go," it said, adding that they had raised the money needed for his treatment.
"We love him more than life itself. If he is still fighting, then we are still fighting."
Gard and Yates exited together after court adjourned. In Gard's breast pocket was little Charlie's favorite stuffed monkey.
On Sunday, Charlie's parents gave the hospital a petition with more than 350,000 signatures from around the world. It called for the family to be allowed to travel for the experimental treatment, although a High Court ruling blocked such a move on June 30.
That ruling was preceded by a series of legal attempts by the couple, who have tried to make their cases to judges in the High Court, the Court of Appeal and the Supreme Court in London. All three British courts have ruled in favor of the hospital's request.
Gard and Yates' legal battle to prolong Charlie's life has received international attention, with Pope Francis and US President Donald Trump
among those weighing in.
US Vice President Mike Pence told "The Rush Limbaugh Show" on Monday that he was praying Charlie's family would "be able to choose the greatest extent of life saving treatment that's available for their child."