After Donald Trump’s election, Lev Parnas saw a chance to cash in on his connections.

The Florida businessman and Trump donor was courting investors for a real-estate project. He took one of them, Roman Nasirov, at the time the head of Ukraine’s tax service, to Mr. Trump’s Florida Mar-a-Lago resort, where they chatted with the president-elect in a crowd of supporters, a video released by Mr. Parnas’s attorney shows. Mr. Nasirov ultimately helped fund a $10 million loan Mr. Parnas arranged for the project, people familiar with...