Harry, 19, died when his motorcycle was hit by a vehicle allegedly driven by Anne Sacoolas, 42, the wife of an American intelligence officer based at RAF Croughton in Northamptonshire on August 27.
The alleged auto driver, Anne Sacoolas, who is married to a US official, subsequently left Britain.
But in a letter from Mr Raab to Mr Dunn's parents Charlotte Charles and Tim Dunn, he said she could now be subjected to a criminal investigation.
On Saturday, Foreign Secretary Dominic Raab wrote to Mr Dunn's family to explain that the British and USA governments now considered Mrs Sacoolas's immunity irrelevant.
Mr Raab added that the matter was now "in the hands" of Northamptonshire Police and the CPS.
The Foreign Office declined to comment. Anne Sacoolas left the United Kingdom under diplomatic immunity despite being a suspect.
"That means that the Americans have made a false claim".
About 23,000 individuals in the United Kingdom have diplomatic immunity, a status reserved for foreign diplomats and their families, as long as they do not have British citizenship. "No loss compares to the death of a child".
"Hopefully we'll get the answers we are seeking", She added.
He added that he was studying the FCO letter "with legal and political experts" to "fully understand where that leaves us".
He also told Sky News it was a "positive step forward", saying: "It's a great sign (Mrs Sacoolas" lawyer has) reached out to us and we will be engaging in embracing those discussions as openly and as positively as we always do'.
Mr Dunn's parents are due to fly out to the USA on Sunday to put pressure on the case.
Sacoolas said in a statement issued on her behalf that she wanted to meet Harry's parents to "express her deepest sympathies and apologies".
"We have been in contact with the family's attorneys and look forward to hearing from them".
"They couldn't speak to her, arrest her, let alone charge her".
It is the first contact between both parties since Mrs Sacoolas left the UK.
"We owe that to the family and obviously this is a terribly tragic case", Patel said when asked if she thought Sacoolas should be extradited to the UK.
Speaking of taking their campaign to the US, Dunn's family said in a statement that they "continue to live in a nightmare" and have so far been unable to grieve after his death. Mr Dunn's father, Tim Dunn, said he was "shocked, but hopeful something can come of this". "Having said that, as it's almost seven weeks now since we lost our boy, sorry just doesn't cut it".