Visiting Jeonnam came into the clash having won each of their last three league outings, but they were blown away by Park's devastating display.
The midfielder opened the scoring in the 11th minute as he reacted first to head the ball home after a cross from the left had evaded everyone and bounced off the post.
He then doubled his tally with a thumping 30-yard strike in the 20th minute, before the visitors halved the deficit through Sim Dong-woon on the half-hour mark.
Park was not done yet, though, and completed his treble with a powerful downward header before making it four in first-half stoppage time with a delightful curling finish from the edge of the area.
Hwang Il-soo added a fifth shortly after the interval and although Stevica Ristic got Jeonnam's second, Thiago rounded off an emphatic victory right at the end.
"It is important for the team, rather than any personal records," Park said.
The crushing victory lifted Jeju above their opponents into fourth in the table, while for Jeonnam it was their heaviest league defeat since July 2012, when they were beaten 6-0 - also by Jeju.
At the top of the table, Jeonbuk Motors moved three points clear of Pohang Steelers with a 2-0 home success over Sangju Sangmu.
First-half goals from Lee Joo-yong and Lee Seung-ki were enough for the leaders, while Sangju's winless streak was extended to five matches.
That victory saw Jeonbuk put the pressure on Pohang, who were unable to gain revenge for their AFC Champions League defeat to Seoul, going down 1-0 at home to the club from the capital as Everton Santos netted an 86th-minute winner.
At the wrong end of the table, bottom club Gyeongnam's woes continued with a 2-1 defeat at Ulsan.
Seo Yong-deok put the home side ahead in the 28th minute, and although Sreten Sretenovic equalised seven minutes after the break Kim Min-gyoon's winner three minutes before the end completed another miserable day for Gyeongnam.
Elsewhere, goals from Hawng Eui-jo and Kim Dong-hee earned Seongnam a 2-0 home win over Incheon United - a first win in seven for Lee Sang-yoon's men.
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