The Republican victory in Virginia’s election for governor is proving particularly influential to party strategists who believe that Glenn Youngkin, a political newcomer, won the Democratic-leaning state in large part by arguing that parents had been shut out of decisions on race in the curriculum, Covid-19 precautions and academic standards.

“If the Virginia results showed us anything, it is that parents are demanding more control and accountability in the classroom,” wrote Rep. Kevin McCarthy (R., Calif.), the House Republican leader, in a letter to colleagues after the election. He said House Republicans would create a “parents bill of rights” soon…

Some Republicans said that the issue had the potential to draw minority voters to the GOP, as well as voters who had moved to the suburbs from urban areas in search of strong school districts. “There’s no issue that resonates like education in the suburbs,” said Christopher Winslow, a Republican on the Board of Supervisors in Chesterfield County, a suburb of Richmond that backed Mr. Biden last year but favored Mr. Youngkin by about 5 percentage points on Tuesday. “That’s where the votes are, and that’s where a Youngkin victory will be heard around the country.”