Last week, a judge decided it was in the best interests of Alfie for his life support to be switched off.
A lawyer representing Tom Evans, 21, and Kate James, 20, asked judges at the Court of Appeal today to rule that Alfie Evans be allowed to leave Britain to receive treatment.
Lord Justcie Davis, Lady Justice Kin and Lord Justice Moylan on Monday ruled against the parents at the hearing in London.
Alfie Evans has been battling an un-diagnosed degenerative brain disease.
They also rejected a request for the case to go to the Supreme Court.
"We want (them) to know they can't break us, we're never going to back down", said Mr Evans.
Alfie Evans, from Liverpool, is on a life support machine and requires round-the-clock treatment at Alder Hey Children's Hospital in the city.
"We will never give up on you Alfie".
Police are now entering Alder Hey hospital as a crowd of protesters gathers outside chanting "save Alfie Evans!"
The parents lost their bid to allow Alfie to be taken overseas for treatment.
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