Charlotte Charles and Tim Dunn turned down an invitation to speak with Anne Sacoolas, who fled the UK following the crash which killed the 19-year-old motorcyclist in August, after the US president told them she was in the next room.
The couple were invited to the Oval Office on Tuesday after travelling to the US in the hope of pressuring Ms Sacoolas to return to Britain to face justice.
Northamptonshire Police have been unable to interview the American, who is married to a US intelligence official, because she claimed diplomatic immunity.
Dunn’s parents travelled to Washington to meet Mr Trump after their lawyer, Radd Seiger, received a call “out of the blue” from the White House on Tuesday.
“I thought we were coming down to have a debate on the diplomatic immunity law. It soon became clear to us that the real reason for inviting us down was to try and get Charlotte and Mrs Sacoolas in a room together,” Mr Seiger told BBC’s Breakfast.
He added: “It was quite an extraordinary meeting. After the pleasantries and greetings, and of course quite sincere and warm condolences, he got down to business and said he had Mrs Sacoolas in the next room.
“I nearly had a heart attack, because I thought he was just going to bring her in. That’s when I quite forcefully said: ‘No, that’s not happening Mr President, these good people are not ready.’”
Mr Seiger said the White House had invited at least three photographers to the meeting and appeared “ready to do a press call”.
Speaking to reporters following the 15-minute meeting with the president, Ms Charles said she was “shocked” when “the bombshell was dropped” that Ms Sacoolas was in the White House.
They turned down the invitation to meet her because they felt it was “not appropriate” without a therapist and mediators present. Mr Dunn said the couple also “wasn’t ready to meet her” and “it would have been too rushed”.
“It’s not what we wanted – we wanted a meeting with her in the UK,” he added.
But US national security adviser Robert O’Brien, who was present for the talks with Mr Trump, told the couple “quite emphatically” that Ms Sacoolas “is never coming back” to Britain, said Mr Seigar.
The lawyer added: “I think the family feel a little ambushed to say the least and disappointed they made the effort to go all the way down there and all the way back up with really no further progress towards achieving the closure they’re so desperately seeking.”
Asked if she felt the meeting was trying to sweep Harry’s death under the carpet, Ms Charles said: “Initially yes, I did think they were trying to do that – certainly by having Mrs Sacoolas there.
“I think maybe they were thinking that would be enough for us. We’re seven weeks on and it’s just not enough.”
She added: “We didn’t get the answers that we wanted. They couldn’t tell us who made the decision to bring her back to the US.”
Ms Sacoolas left the UK days after the 27 August crash which killed Dunn near RAF Croughton. She later confirmed she had been driving on the wrong side of the road when her 4x4 smashed into the teenager’s bike and said she was “terribly sorry”.
Mr Dunn said her statement had made the situation “a little bit worse”.
“Now she’s admitted it, we don’t understand why she’s not coming back,” he told BBC Radio 4’s Today. “She’s admitted her guilt but we still can’t get any further forward with getting her back for some justice for Harry.”
Northamptonshire Police was wrongly advised by the UK Foreign Office that Ms Sacoolas had diplomatic immunity, allowing her to return to the US without being interviewed.
Mr Seiger said Dunn’s family were considering legal action to establish “whether somebody cocked up or whether they were put under pressure by the Americans”.
The teenager’s mother told ITV’s Good Morning Britain on Wednesday she did not hold much hope that Ms Sacoolas would voluntarily return to the UK.
“I think we’re just going to just be forever disappointed, forever disgusted in both the UK and US governments. It just doesn’t make sense to us as to why on earth this should be allowed to happen,” she said.
The White House has been contacted for comment.