Nearly four decades ago, a group of lawyers, intellectuals and activists assembled in a hotel ballroom in Taipei to found an illegal political party dedicated to ending authoritarian rule in Taiwan.

No longer a scrappy upstart, the Democratic Progressive Party, born in that ballroom, is now seeking an unprecedented third consecutive term. It needs to persuade voters that after eight years in power, the party can renew itself while also protecting Taiwan from mounting pressures imposed by Beijing, which claims the island as its territory.

Led by Vice President Lai Ching-te, the presidential candidate, the D.P.P. faces a stiff challenge in an election on Saturday from its chief rival, the Nationalist Party, which favors expanded ties with China. Polls have indicated that the Nationalists, led by Hou Yu-ih, a former policeman and the mayor of New Taipei City, may have a fighting chance of returning to power for the first time since 2016, an outcome that could reshape the region’s geopolitical landscape. Election results are expected by Saturday night.

For Su Chiao-hui, a lawmaker with the Democratic Progressive Party, the stakes of the vote are especially personal. Her father, Su Tseng-chang, helped found the party when Taiwan was under martial law and later served as a premier in both the party’s two phases in power, including under the current president, Tsai Ing-wen.