SEOUL, South Korea — President Park Geun-hye of South Korea on Tuesday asked rival political parties to recommend a new prime minister, making a major political concession in her efforts to defuse an influence-peddling scandal that has crippled her government for weeks.

During a meeting with Chung Sye-kyun, the speaker of the National Assembly, Ms. Park made the proposal, essentially retracting the appointment, made six days ago, of her own candidate for prime minister. That candidate, Kim Byung-joon, was rejected by the opposition parties who together control a majority in Parliament.

Instead, they demanded that Ms. Park cede some of her presidential policy-making power, especially in economic and other domestic issues, to a prime minister recommended by Parliament. (In the meantime, Prime Minister Hwang Kyo-ahn, whom Ms. Park had wanted to replace with Mr. Kim, remains in office.)

The concession was another political blow to Ms. Park, and new evidence of her much-diminished political authority as the scandal continues to unfold. She has tried to regain public trust by reshuffling her cabinet and presidential staff.