With Mr. Trump, there was no special prosecutor investigating the Ukraine matter so it was left to the House itself to unearth the details of what happened. But the president refused to turn over documents and tried to block testimony by current and former advisers. That led Democrats to make the strategic decision not to wait for a prolonged court fight to force key witnesses like Mr. Bolton to testify, reasoning that the evidence they had already turned up was enough to justify articles of impeachment. But they said that decision should not stop the Senate from trying to get to the truth.

In the Clinton case, the fight focused on witnesses who had already testified during Mr. Starr’s grand jury and they had no new information to provide during the Senate trial. Refusing to hear from Mr. Bolton or others who have never testified like Mick Mulvaney, the acting White House chief of staff, means senators would decide guilt or innocence without access to the fullest version of the facts.

After complaining in the House that the witnesses who testified generally offered secondhand or hearsay accounts, Republicans would now be in the position of turning down testimony from advisers who do have firsthand information.

“It seems evident the Senate will have to call witnesses if they are going to uncover the rest of the story,” said Byron L. Dorgan, a Democrat from North Dakota who was a senator during the Clinton trial.

Still, there is risk involved for Democrats prosecuting the president. Mr. Parnas in some ways mainly amplifies what is already known from other evidence and to the extent that he adds to the case against Mr. Trump, his credibility could be attacked given that he has been indicted on campaign finance charges.

As for Mr. Bolton, no one knows for sure what he would say if he did testify. While he was described as critical of the Ukraine pressure campaign by other officials, it is not known whether he would implicate or exonerate the president himself. He left the White House on acrimonious terms and has criticized some of the president’s foreign policy decisions, but he has not become a Never Trumper-style critic and some Democrats are privately nervous about his potential testimony.

Even if they do not end up with the witnesses and documents they want, Democrats argue that the latest revelations from Mr. Parnas and the G.A.O. report indicate that the House charges were on track.