Chris Gard and Connie Yates want Mr Justice Francis to rule that 11-month-old Charlie, who suffers from a rare genetic condition and has brain damage, should be allowed to undergo a therapy trial in the United States.
Specialists at Great Ormond Street Hospital in London, where Charlie is being cared for, say the therapy is experimental and will not help.
They say life-support treatment should stop.
The couple, who are in their 30s and come from Bedfont, west London, are mounting the latest stage of their fight at a hearing in the Family Division of the High Court.
Charlie's parents have already lost battles in the High Court, Court of Appeal and Supreme Court in London.
They have also failed to persuade European Court of Human Rights judges to intervene.
The couple now want Mr Justice Francis to carry out a fresh analysis of their case.
The couple walked out about two hours into Thursday's hearing.
Mr Gard stood up and said: "I thought this was supposed to be independent."
Mr Justice Francis had earlier told the hearing: "If there is important evidence which suggests that I should change my decision then I will change it."
Barrister Grant Armstrong, who is leading Charlie's parents' legal team, indicated the couple thought they had evidence which was new.
He said the plan was for a specialist in the US who was offering therapy to give evidence at the hearing via a link from America.
Mr Armstrong said a hospital in Rome had also made an offer.
And he said there was evidence to suggest the proposed therapy would not be futile.
Mr Armstrong said the American doctor offering treatment was a "world authority".
He said the therapy on offer was not "fringe science".
"There is a respectable body of authoritative opinion," he said.
"There are treatments available."
Mr Armstrong said: "The parents seek to re-open the case in relation to the chances of success of treatment."
He said Great Ormond Street doctors had agreed to keep providing life-support treatment pending the outcome of the latest stage of litigation.
Mr Armstrong said Charlie's parents' unease at decisions made by judges is shared by members of the public.
"This case does raise some important issues," he said.
"There is a proper debate from those who are informed."
Mr Justice Francis said he was unlikely to make a "final determination" on Thursday.