The family of Harry Dunn have begun their legal proceeding against the Foreign and Commonwealth Office (FCO).
Mr Dunn, 19, died in a crash in Northamptonshire in August which led to the suspect, Anne Sacoolas, leaving for the USA under diplomatic immunity.
His parents have filed a claim for judicial review in the High Court in London against Foreign Secretary Dominic Raab.
Family spokesman Radd Seiger said the FCO was “wrong in law”.
He said: “The parents have done everything physically within their power to avoid having to sue the FCO.
“There have been repeated public and private attempts on our part to engage with those in authority to resolve this dispute amicably.”
Mr Seiger said the family’s legal team has advised them “the FCO’s interpretation of the diplomatic immunity laws and treaty is absurd in practice and wrong in law”.
The FCO said last month the family had not found “any reasonably arguable ground of legal challenge”.
It said an allegation that Mr Raab had misused or abused his power was “entirely without foundation”.
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The family said that a letter sent to them from the foreign secretary confirmed that the FCO would “seek full costs” for the legal challenge.
Mr Raab had previously said this was to “protect taxpayers’ money”.
The family said the cost of the case could be “upwards of £50,000”.
A spokesman for the FCO said: “We have deep sympathy for Harry’s family. We have done and will continue to do everything we properly can to ensure that justice is done.
“As the foreign secretary set out in Parliament, the individual involved had diplomatic immunity whilst in the country under the Vienna Convention on Diplomatic Relations.”
Mr Dunn was fatally injured on 27 August, when his motorbike was in collision with a car owned by Anne Sacoolas outside RAF Croughton, where her husband Jonathan was an intelligence officer.
Mrs Sacoolas, 42, left the UK claiming diplomatic immunity.
On Monday, Harry Dunn’s father Tim Dunn attempted to speak to Mr Rabb at his constituency hustings in Surrey.
A member of staff at the church where the hustings were held said he was kept outside due to fire safety.