President Park Geun-hye's visit to a rural market came as opponents in parliament squabble over whether to impeach her.

Police said they arrested a 48-year-old man who told investigators he used paint thinner to set fire to the birthplace of Park's late father on Thursday because he was angry over her refusal to step down immediately over the scandal involving a confidant accused of influence peddling and extortion.

A memorial hall for Park's parents, ex-President Park Chung-hee and Yook Young-soo, was nearly burned down. The hall is a popular tourist attraction. The suspect was previously convicted of setting fire to the birthplace of another former president, Roh Tae-woo, in 2012, according to police.

Park didn't immediately comment publicly on the arson. Earlier Thursday, she visited a market in the nearby city of Daegu where hundreds of shops had been destroyed by a previous fire.

Daegu is Park's political home turf where she was elected as a national lawmaker four times before becoming president in 2013.

Local TV showed dozens of people at the market shouting "Park Geun-hye! Park Geun-hye!" and using cellphones to photograph her as she walked through the fire-ravaged shops. One middle-aged woman wiped away tears. Near the market, though, dozens of anti-Park citizens staged rallies calling for her ouster, according to media reports.

"Merchants [here] supported me whenever I had difficulties and I'm really sorry," Park said during her visit, according to her office. "I agonized a lot over whether I should come here in the current situation."

Park cried after returning to her car, her office said, citing presidential security staff who traveled with her.

On Tuesday, Park said she would step down if parliament arranges a safe transfer of power, drawing fierce criticism from main opposition parties that described her overture as a tactic to buy time that would allow her to survive the scandal.

Park's offer appears to have caused cracks among dissenters in her ruling party who earlier supported her impeachment. A group of anti-Park lawmakers is now saying they won't join an impeachment drive if Park resigns in April and helps ensure a stable power transfer until a new president takes office following a presidential by-election.

A meeting between leaders of the main opposition parties on Thursday ended without any major breakthrough. They differ over whether they should push for Park's impeachment this week or delay it.

Much of the hesitation over Park's impeachment is because the opposition does not have enough lawmakers to pass an impeachment in parliament, and they would need help from dissenters in Park's ruling Saenuri Party.

Passage of an impeachment motion requires at least 200 votes in the 300-member National Assembly, but the three opposition parties and anti-Park independent lawmakers have a total of 172 seats.

If impeached, Park's presidential powers would be suspended until the Constitutional Court rules on her fate. The court would have 180 days to deliberate.

Park has denied accusations by prosecutors that she colluded in the criminal activities of her longtime friend Choi Soon-sil, who, despite having no official role in government, allegedly had a say in policy decisions and exploited her presidential ties to bully companies into giving millions of dollars to businesses Choi controlled.

Choi and two of Park's former advisers have been detained and indicted on charges including extortion and leaking confidential information.

Choi is a daughter of a cult leader who served as Park's personal mentor after her mother's assassination in 1974. Park's father was gunned down by his spy chief in 1979 following an 18-year rule.

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