Charlotte Charles and Tim Dunn told the PA news agency they had given the Government a chance to “come clean to us”, but claimed they were “slapped in the face each time”.
Acting on behalf of the 19-year-old’s parents, their spokesman Radd Seiger called for the inquiry to investigate who decided suspect Anne Sacoolas could leave the UK and “who ordered that the matter should be covered up”.
He suggested an inquiry should consider a replacement for the “imbalanced” extradition treaty between the UK and the US and to look into why an ambulance took more than 40 minutes to get to Mr Dunn.
PA understands a letter was previously written to Northamptonshire Coroner Anne Pember over the possibility of an inquest being turned into a full public inquiry – a request which has been referred to Chief Coroner Mark Lucraft QC.
Mr Dunn was killed when his motorbike crashed into a car outside a US military base in the county on August 27 last year.
Sacoolas, 42, the wife of a US intelligence official, claimed diplomatic immunity following the crash and was able to return to her home country.
She was charged with causing death by dangerous driving in December but an extradition request submitted by the Home Office for the suspect was rejected by US secretary of state Mike Pompeo in January – a decision the State Department have since described as “final”.
After Mr Seiger called for an inquiry on their behalf, Charlotte Charles and Tim Dunn told PA: “We can’t bring Harry back. We know that.
“But like anyone else, we are entitled to see that justice is done for our son, who did nothing wrong.
“Despite what the Government think, his life did matter and we are so grateful to Radd for knocking on our door that day when we had been told we had no chance of getting justice.”
The teenager’s parents continued: “Then with the help of the media, the whole nation and millions of people all around the world, including the wonderful people of America, came to support us too. That has meant the world.
“Without public pressure, this whole disgusting saga would have been buried. We promised Harry we would get him justice.
“That is what we will do and no-one will get in our way.
“We have given the Government every chance to come clean to us. They slapped us in the face each time, clearly trying to hide what they did.
“Our only wish now, apart from ensuring Mrs Sacoolas faces justice, is that this should never happen to another family again.
“Following our experience, this Government clearly cannot be trusted to look after us, the people.
“As a nation, we have to get back to a more decent way of doing things.
“This is not the sort of society we should be where our elected officials think it’s OK to kick those in most need of help.”
Calling for the inquiry on behalf of the parents, Mr Seiger told PA: “Given the horrific nature of what has happened in this case, with all its different strands and the extent to which Government has sought to cover it up, there must be an inquiry the nation can have confidence in.
“An inquiry can restore trust and faith in Government, so as to ensure that anyone involved in misconduct in public office are held to account and so that lessons are learned and systems are put in place so that no such abuse of human rights is committed by Government again.”
A spokeswoman for the Foreign Office said: “We have deep sympathy for Harry’s family. We are confident that we have acted properly and lawfully in relation to Harry’s death.
“Both the Prime Minister and the Foreign Secretary have been clear with the US that the refusal to extradite her (Sacoolas) amounts to a denial of justice, and that she should return to the UK.”