Chris Gard and Connie Yates, from west London, want 10-month-old Charlie to take part in a therapy trial in the United States.
Specialists at Great Ormond Street Hospital think he has no chance of survival and want to end his life support treatment.
Judges in the European Court of Human Rights are starting to analyse the case of a terminally-ill baby who has been at the centre of a high-profile legal battle in London.
Specialists at Great Ormond Street Hospital for Children in London, where Charlie is being cared for, say life support treatment should now be switched off and argue the therapy proposed by a doctor in the USA is experimental and will not help.
They say life support treatment should stop and it would be "unethical" to keep him alive as he could be suffering.
Charlie's parents have asked European court judges in Strasbourg, France, to consider their claims after losing battles in the High Court, Court of Appeal and Supreme Court in London.
"In light of the exceptional circumstances of this case, the court has already accorded it priority and will treat the application with the utmost urgency", she added.
Charlie's parents have exhausted legal options in the UK.
A High Court judge ruled against Charlie's family in April, a decision backed by the Court of Appeal and the Supreme Court.
Lady Hale said Mr Justice Francis had decided what was in Charlie's best interests.
He heard that Charlie, who was born on August 4 previous year, had a form of mitochondrial disease, a condition which causes progressive muscle weakness and brain damage.
He has previously argued parents should be free to make decisions about their children's treatment unless any proposed care poses a risk of significant harm.
Barrister James Eadie QC, who led the Government's legal team, said he would argue judges in London made decisions they were entitled to make and human rights legislation had not been breached.
"We respectfully urge our colleagues in the European Court of Human Rights to do everything in their power to address the proposed application by then".