Tom Evans, 21, and Kate James, 20, had asked Court of Appeal judges to rule that 23-month-old Alfie could go overseas for treatment.
Judges have heard how Alfie, born in May 2016, is in a "semi-vegetative state" and has a degenerative neurological condition that doctors have been unable to diagnose definitively.
Court of appeal judges ruled yesterday that Tom Evans, 21, and Kate James, 20, could not take their son Alfie Evans overseas to receive treatment for a rare degenerative brain disease.
Protesters from "Alfie's Army" have lined the roads outside the specialist children's hospital in support of the latest court case.
"We fully understand what a sensitive and emotional time this is for everyone involved and I would also therefore like to pass on our appreciation for the way in which Alfie's family were later able to speak to the crowd and offer assurance and calm".
Most of the protesters have now moved from outside the hospital's main entrance to Springfield Park, a short distance away.
Last month, Mr Justice Hayden had ruled that it was in Alfie's best interests for his treatment to be withdrawn.
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Supreme Court justices and European Court of Human Rights judges refused to intervene.
Barrister for the parents, Paul Diamond, said the couple might make a further appeal to the Supreme Court.
After the High Court judge upheld his original December 2017 ruling that backed Alder Hey doctors in their decision to switch off the toddler's life support, on Thursday night, supporters gathered at the hospital to protest.
Alfie's parents say their son has improved in recent weeks and had asked Mr Justice Hayden to allow a new assessment.
Alfie's parents today thanked "thousands" for their "historic show of support" outside Alder Hey Children's Hospital Trust in Liverpool and called on them to return.
According to weekend news reports, Alfie Evans' mother was told to leave his bedside after the hospital informed Alfie's parents that they would no longer be able to sleep in his hospital room overnight. Moylan argued, however, that both Bambino Gesu staff and staff at Alder Hey agree that no hope remains for curative or preventative treatment for Alfie and that he should receive end of life care.
Evans' mention of the hospital not allowing the family to be together was in reference to a letter he shared last week in which Alder Hey accused pro-Alfie protesters of causing "significant disruption" to other patients' families and staff.
"The bad reality was that nearly the entirety of Alfie's brain has been eroded, leaving only water and cerebral spinal fluid", Moylan said, reading from Hayden's previous decision, according to The Sun.