“If he gets a parliament that is not dominated by anti-Rouhani people, he is going to have a much easier time to push through some of the changes he wants,” said Trita Parsi, who heads the National Iranian American Council.

At the same time, Mr. Parsi said, if Rouhani supporters come away with a majority in the 290-member “Majlis,” or parliament, there will be “greater pressure on him to do what he promised to do originally on improving civil and human rights in Iran.”