And they said they turned down the chance to meet the diplomat’s wife, who has reportedly admitted to driving on the wrong side of the road, after being told she was in the next room in the White House.
Radd Seiger, a lawyer representing Harry's parents, said: "It was almost like, 'Hey, I've got someone who wants to see you.'
"Then I looked to my side and I saw at least three photographers ready to do a press call."
He added: "The family feel a little ambushed to say the least and disappointed they made the effort to go all the way there and all the way back with little progress towards achieving the closure they are so desperately seeking."
Mr Dunn said: “We weren't ready to meet her, it would have been too rushed.
"It's not what we wanted – we wanted a meeting with her in the UK."
But he added that the meeting "didn't feel like a stunt – they didn't try and force it on to us”.
Mrs Charles said: "I don't think it would be appropriate to meet her without therapists or mediators in the room."
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Harry’s parents described the President as "very graceful" but said they are no "further along" after being told that Mrs Sacoolas would not return to the UK.
Mr Dunn said: "I think the President was very graceful and spoke very well to us.
"Obviously we've just met the President and we never thought we'd get this far, but I don't know whether we're any further along.”
Mrs Charles said: "When [President Trump] held my hand, I gripped it a lot tighter and I was honest with him and just said, as I said a while ago, ‘If it was your son you would be doing the same as us.’
"He actually gripped my hand a little bit tighter and said: 'Yes, I would be.’
"And that's when he said he would try and look at this from a different angle.
"He seemed to understand that our grief has been locked in for seven weeks now on a case that should have been very, very clear-cut.”
Mr Dunn said: "He listened to Charlotte very well – she spoke excellently to him and he was very understanding.
"I genuinely do think he will look to resolve this in a way that will help us.”
However, Mrs Charles said they “didn’t get the answers we wanted”, adding: "They couldn't tell us who made the decision to bring her back to the US.
"We asked how long she was there for and they still said three weeks.”
Mrs Sacoolas has previously said she is "terribly sorry" about the incident and that she had "no time to react" when she saw 19-year-old Harry’s motorbike following the incident near RAF Croughton on 27 August.
Trump previously said that driving the wrong way “happens” when asked to comment on Harry’s death.