Chris Gard and Connie Yates had been expecting their 10-month-old's life support to be turned off on Friday.
Great Ormond Street Hospital has since disclosed it is putting plans in place for Charlie's care to allow his family to spend more time with him.
On Tuesday, Charlie's parents lost their final legal appeal to take him to the US for experimental treatment.
Judges at the European Court of Human Rights concluded that further treatment would "continue to cause Charlie significant harm", in line with advice from specialists at Great Ormond Street.
He has a rare genetic disease as well as brain damage and is believed to be one of 16 children in the world to have the condition; mitochondrial depletion syndrome.
Doctors have said he cannot see, hear, move, cry or swallow.
Charlie has been receiving specialist treatment at Great Ormond Street Hospital since October 2016.
His parents said they had been denied their final wish to be able to take their son home to die and felt "let down" following the lengthy legal battle.
Alongside a video posted on YouTube on Thursday, Charlie's parents wrote: "We are utterly heartbroken spending our last precious hours with our baby boy.
"We're not allowed to choose if our son lives and we're not allowed to choose when or where Charlie dies.
"We, and most importantly Charlie, have been massively let down throughout this whole process."
A spokesperson for Great Ormond Street Hospital said earlier: "As with all of our patients we are not able to, and nor will we, discuss these specific details of care.
"This is a very distressing situation for Charlie's parents and all the staff involved and our focus remains with them."
Charlie's parents raised Â£1.3m on a crowdfunding site to pay for the experimental treatment in the US.
Ms Yates previously indicated the money would go towards a charity for mitochondrial depletion syndrome if Charlie did "not get his chance".
Charlie Gard's parents have been told they will be able to spend more time with their terminally ill baby.