Tom Evans, 21, and Katie James, 20, said sorry after a protest outside Alder Hey Children's Hospital allegedly intimidated doctors and nurses and threatened to cut short their precious last moments with the 23-month-old tot.
Barrister Paul Diamond said on Monday that there was a "passion out there" about Alfie Evans' case.
Specialists at Alder Hey said life-support treatment should stop and Mr Justice Hayden said he accepted medical evidence which showed that further treatment was futile.
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 outside Alder Hey Children’s Hospital in Liverpool
Alfie, from Bootle, Merseyside, has been at Alder Hey since December 2016 with an undiagnosed degenerative brain disease.
"Our understanding of the law is that they do, that Alfie's not a ward of the state, that means he's not a prisoner of the hospital, and that the parental rights of Kate and Tom have not been extinguished", Kiska said.
"We want (them) to know they can't break us, we're never going to back down", said Mr Evans.
"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".
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On the first occasion Marko Arnautovic was offside when he headed home - and then he was interfering with play from Edimilson Fernandes' shot.
Those hearings led to Mr Justice Hayden setting a time and date for life support to be switched off at a High Court Hearing earlier this week.
Justice Moylan denied permission to appeal to the Supreme Court.
Alder Hey Children's Hospital said in a statement: "At each stage of the legal process the courts have agreed with expert advisers... that Alfie's condition is untreatable".
They lost a challenge at the Court of Appeal to the ruling he should have palliative care and failed to have the decision overturned at the Supreme Court and European Court of Human Rights.
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Alfie's parents say their son has improved in recent weeks and had asked Mr Justice Hayden to allow a new assessment.
The letter also stated that family members and friends of the parents were last night "shouting over your son Alfie in his cubicle, abusive, filming other patients and visitors, filming staff, and behaving in a manner which was felt to be threatening, intimidating and unsafe by other parents and by our staff".
Hayden said details of that plan could not be revealed because Alfie was entitled to privacy at the end of his life.
However, Alder Hey said in a statement: "Noise from recent protests has unfortunately affected our patients so we would ask that noise levels outside the hospital are kept to a minimum and, for example, auto horns are not sounded".
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Police said the "large" and "peaceful" protest in Liverpool on Thursday night "did cause significant traffic disruption and inconvenience for other people trying to access the hospital".
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