The pair carded rounds of 68 and 69 respectively to move to nine-under for the tournament, and hold a slender advantage over second round leader Brooke Henderson, Karrie Webb and Angela Stanford.
Park bounced back from a bogey at the second to register birdies at the fourth, seventh and 14th holes, before a bogey at the 17th was cancelled out by a fourth birdie at the last.
Thompson started strongly, birding the first and picking up further shots at the seventh and 10th holes. A bogey at the 11th would prove her lone blemish of the day, with a further biride on 18 ensuring the American finished strongly.
It was Webb who shot the low round of the day, with the Australian's 64 containing nine birdies – six of which were on the front nine – and two bogeys.
Cristie Kerr and Hee Young Park are a further shot adrift on seven-under.
World number one Lydia Ko is some nine shots off the lead on even par, having carded a third round 70.