Archie Battersbee's parents have launched a new attempt to stop their son's life support being turned off.
Archie, 12, has been in a coma since 7 April after being found unconscious by his mother having suffered a catastrophic brain injury at his home in Southend, Essex.
She believes he was taking part in an online challenge.
His life support was due to be switched off on Tuesday afternoon, the day after the Court of Appeal rejected a bid to postpone the ending of his treatment.
His parents, Paul Battersbee and Hollie Dance, have now appealed to the Supreme Court to allow more time for a complaint they made to the UN Committee on the Rights of Persons with Disabilities to be considered.
Last week the pair made an application to the UN, which issued a request that the UK Government “refrain from withdrawing” his treatment while it considered the complaint.
However, three senior judges at the Court of Appeal hearing on Monday refused to further extend the stay on withdrawing Archie’s treatment
A spokesman for the Christian Legal Centre, which is supporting the legal action brought by Archie’s parents, said the hospital trust has confirmed it will not take any steps to withdraw treatment until the Supreme Court has reached a decision.
Watch: Archie Battersbee - timeline of 12-year-old's case so far
Ms Dance said: “We are having to battle over every decision with the hospital.
“There is nothing dignified in how we are being treated as a family in this situation.
“We do not understand what the rush is and why all of our wishes are being denied.
“I know Archie’s still with us. Archie’s showing very different signs to what the clinicians are actually putting over to the courts.
“He’s very much there, he’s progressing in so many ways.
“We pray for an encouraging response from the Supreme Court.”
Doctors who are caring for Archie have said they believe he is brain-stem dead, and argue it is in his best interest for life-support measures to be withdrawn.
The High Court has also ruled that Archie's "every bodily function is now maintained by artificial means".
However his mother has asked for more time after she said she has seen signs "of progression".
She told Good Morning Britain on Tuesday morning: "Archie is showing very different signs. He’s progressing in so many ways.
"He regulates his own body temperature, he has a stable heart beat, he holds his own blood pressure, he’s gaining weight, all different things that haven't been put over.
"Archie has held my hand, he’s held other people’s hands, squeezed fingers, he squeezed them so tight my fingers were red.
"He’s opened his eyes, he’s attempted to breathe, all we’re asking for is time."
Archie’s care was initially due to end at 2pm at the Royal London Hospital in Whitechapel on Monday.