The same scandal saw prosecutors carry out a morning swoop on the offices of South Korea's largest conglomerate Samsung Electronics, looking for incriminating documents.
In a significant political concession, Park told the speaker of the National Assembly she would accept a prime minister chosen by the opposition-controlled legislature "and let him control the cabinet".
The premiership is normally a largely symbolic post in South Korea, where power is firmly concentrated on the executive.
Her climbdown underlined just how weakened she has been by the scandal involving a close personal friend, Choi Soon-Sil, who has been arrested on charges of fraud and abuse of power.
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The charges relate to allegations that Choi leveraged her personal relationship with Park to coerce donations from large companies like Samsung to non-profit foundations which she set up and used for personal gain.
She is also accused of interfering in government affairs, including the nomination of senior officials.
In a bid to restore public trust, Park reshuffled her advisers and senior cabinet members, and nominated a liberal candidate for prime minister from outside her conservative Saenuri Party.
But opposition parties had vowed to block her nominee on the grounds they were not properly consulted.
During their meeting, parliament speaker Chung Sye-Kyun told Park her biggest priority should be to alleviate widespread public concern and anxiety.
"The president's crisis is the crisis of state affairs and the crisis of the nation. If the National Assembly recommends a nominee, you must appoint him and grant him authority and ensure there will be no disputes in the future about his authority," Chung said, according to an official transcript.