The ship’s captain, Lee Jun-seok, 69, and 14 other top members of the crew escaped the ferry in the first two coast guard ships to arrive at the scene. All of them are now under arrest on criminal charges, including accidental homicide.

The prime minister is a largely ceremonial post in South Korea, with the executive power concentrated in the president, and is sometimes fired when the government needs to soothe public anger after a major scandal or policy failure.

Ms. Park’s tumultuous first year in office has been rocked by scandals in domestic politics, as well as unusually high tensions with North Korea following its nuclear test early last year.

Under Ms. Park, the government has initiated a campaign to eliminate what it called “jongbuk,” or followers of the North Korean government, from South Korean society and its politics, asking the Constitutional Court to disband a small progressive party on charges of acting on the Communist North’s behalf. The political opposition accused Ms. Park of resorting to a Communist witch hunt — a tactic her father, the former military strongman Park Chung-hee, was accused of using — to maintain her sway over domestic politics.

Prosecutors revealed last month that government agents fabricated evidence to build a spy case against a defector from North Korea. Separately, a former chief and senior officials at the government’s National Intelligence Service have been on trial on charges of directing an extensive online smear campaign against Ms. Park’s opponents ahead of her election in December 2012. They conducted the online campaign in the name of fighting pro-North activities on the web.

No evidence has emerged that Ms. Park was involved in any of these scandals.

“It’s utterly irresponsible and cowardly” for the prime minister to step down in the middle of the rescue operation, Ahn Cheol-soo, a top leader of the main opposition party, New Politics Alliance for Democracy, said during a news conference on Sunday.

Before firing Mr. Chung, Ms. Park herself should offer an apology, Mr. Ahn said.

When she took office in February last year, one of the first things Ms. Park did was to change the name of the old Ministry of Public Administration and Security to the Ministry of Security and Public Administration. By putting “security” up front, Ms. Park was stressing that her government was dedicated to securing the safety of the people, officials said at the time.