Tom Evans and Kate James wanted Alfie to be transported to a hospital in Rome for care, but the Court of Appeal upheld a ruling this week preventing the 23-month-old from travelling overseas after life support was withdrawn.
The news report came out after Tom Evans told the media that he would be working with Alder Hey Children's Hospital in an attempt to fix the relationship so Alfie can go home.
Doctors have been unable to successfully diagnose Alfie Evans, but he is thought to have an incurable neurological condition.
The boy's parents have been fighting one court battle after another for the right to take Alfie to Italy for care at a children's hospital there.
The British hospital treating Alfie withdrew his life support Monday after a series of court rulings sided with the doctors and blocked further medical treatment.
Earlier today, Tom Evans said there had been no deterioration in Alfie's condition since he was taken off a ventilator and he was not in pain, accusing doctors of misdiagnosing his son.
"Our lives have been turned upside down by the intense focus on Alfie and his situation", Evans said outside Liverpool's Alder Hey Hospital, where Alfie has been treated for more than a year.
Alfie's mother, Kate James, shared a video Wednesday on Facebook after the dismissal of the appeal. "Your holiness, save our son", he told the Pope. He's been outspoken about the parents' rights in Alfie's case and believes they and others have "moral rights" when it comes to the healthcare of their children.
Emotions have run high over the case, with a band of supporters known as "Alfie's Army" protesting regularly outside the hospital, at times trying to storm the entrance.
Think tank Parliament Street has now launched a campaign with a press conference in Westminster in bid to get MPs to debate giving parents more power over children's hospital treatment.
'I would like to make people aware that these posts are being monitored and remind social media users that any offences including malicious communications and threatening behaviour will be investigated and where necessary will be acted upon'.
The boy's parents insisted that it is in their son's best interest to travel to Rome, where Alfie was granted Italian citizenship, to seek additional treatment.
But the archbishop added: "I am very aware of the compassion which is characteristically shown by the Italian people to those in need, and in this case Alfie".
"It also means that we owe it to Alfie to have the courage to do what is right for him now that the best course has been determined".
Polish President Andrzej Duda has also lent his support, saying in a Tweet: "Alfie Evans must be saved!"
"His courageous little body has proved again that the miracle of life can be stronger than death", the president wrote on Twitter.
Doctors linked with Pope Francis had been ready to fly Alfie to Rome in a military air ambulance that was put on standby outside Alder Hey.
They added: "Unfortunately, these same remarkable staff have recently been the target of unprecedented personal abuse that has been hard to bear".
Cruz said the events "remind us of the tragic case of Charlie Gard past year".
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