Mr Dunn, 19, was killed when his motorbike crashed into a car being driven on the wrong side of the road by American Anne Sacoolas outside RAF Croughton in Northamptonshire on 27 August last year.
Ms Sacoolas, whose husband Jonathan Sacoolas worked as a technical assistant at the base, left the country a few weeks later after the US said she was entitled to diplomatic immunity.
The 43-year-old was ultimately charged with causing death by dangerous driving last December, but an extradition request was rejected by the US State Department in January - a decision it later described as "final".
Mr Dunn's parents, Charlotte Charles and Tim Dunn, claimed the Foreign, Commonwealth and Development Office (FCDO) wrongly decided Ms Sacoolas had diplomatic immunity and unlawfully obstructed Northamptonshire Police's investigation into their son's death by keeping the force "in the dark".
The judges also rejected Ms Charles and Mr Dunn's claim the FCDO "usurped" Northamptonshire Police's investigation into their son's death, finding officials "sought to assist rather than obstruct Northamptonshire Police in their investigation".
But, in a High Court judgment delivered on Tuesday, Lord Justice Flaux and Mr Justice Saini said: "Our conclusion is that Mrs Sacoolas enjoyed immunity from UK criminal jurisdiction at the time of Harry's death."
Speaking to the PA news agency after the ruling, Ms Charles said the court ruling was “just a blip along the way”.
She added: "I promised my boy I would get him justice and that is just what we are going to do. No one is going to stand in our way.
"It's obviously disappointing that this court did not find in our favour but we are more focused now than ever on fulfilling our promise."
At a hearing earlier this month, their lawyers said the FCDO "took upon itself the authority to resolve the question of immunity and ultimately and unlawfully decided to accept the US embassy's decision that Anne Sacoolas had immunity".
Sam Wordsworth QC told the court that Ms Sacoolas had "no duties at all" at the base and therefore "never had any relevant immunity for the US to waive".
But Lord Justice Flaux and Mr Justice Saini found Ms Sacoolas had diplomatic immunity "on arrival in the UK" under the Vienna Convention on Diplomatic Relations (VCDR) which had not been "expressly waived", meaning she "had immunity at the time of Harry's death".
The judges said: "We do not come this conclusion with any enthusiasm for the result, but it is compelled by the operation of the VCDR."
Mr Dunn’s father said he “still wakes up every morning in absolute disbelief that we are in this situation at all”.
He added: "It's bad enough feeling the horrible pain of not having Harry around and missing him, but I can't believe the governments are putting us through this.
"It all seems so cruel and needless and I am just as angry today as I ever have been but so determined to see it all through until we have justice."
The Foreign Secretary, Dominic Raab, said in a statement after the hearing: “My thoughts today are with Harry’s family. While this judgment makes clear the Foreign Office acted properly and lawfully throughout, I appreciate that won’t provide any solace to the family in their search for justice.
“We stand with them, we’re clear that Anne Sacoolas needs to face justice in the UK, and we will support the family with their legal claim in the US.”
Downing Street said an agreement had been reached to ensure that Mr Dunn’s family “will be protected from having to pay high costs to the government in respect of the High Court proceedings”.
Additional reporting by PA