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".

The Swiss will turn 35 during the Rio Games, where he will play in the singles and mixed doubles, although he has not decided on the men's doubles.
"I've planned all of 2016, you know all the way through the Rio Olympics and beyond," he said.
"No plans to retire yet. I don't have a definite date even though that would help to make things easier to plan."
Federer, who has been competing in the International Premier Tennis League in India, added: "I'm looking forward to next year.
"The Australian Open is obviously a big goal for me. And after that it's going to be a long, tough year.
"So I'm happy I'm feeling fine physically and in good shape and of course, like I say so many times I hope I'm still on tour for a while."

The engine was driven from Larne fire station at about 04:00 GMT on Saturday and crashed into cars and houses a short time later.
Robert Duffin, 66, from Green Link in Larne, appeared at Coleraine Magistrates' Court on Monday facing five charges linked to the incident.
Ross Clarke, 19, from Fairway in Larne, also appeared, accused of 14 offences.
He faces charges including burglary, aggravated vehicle taking, causing damage and various motoring offences.
The pair were remanded in custody to appear again on Thursday at Ballymena Magistrates' Court, when it is expected a bail application will be made.
Charlie Gard's parents have been told they will be able to spend more time with their terminally ill baby.
[NEXT_CONCEPT]
Roger Federer, the 17-time Grand Slam singles champion, does not intend to retire after next year's Olympics.
[NEXT_CONCEPT]
Two men have been remanded in custody over the theft of a fire engine in County Antrim.