DALLAS, 2 May 2015 (AFP) - Thompson also had four birdies and just one bogey in a 68 as she joined the South Korean star on nine-under 204 after 54 holes.
Canadian 17-year-old Brooke Henderson, who held the first-round lead after qualifying for the field on Monday, had at least a share of the lead much of the day, but faded with two late bogeys in an even par 71.
That left her tied for third on eight-under 205, where she was joined by Australian Hall of Famer Karrie Webb (64) and American Angela Stanford (67).
Park shrugged off a bogey at the second with birdies at four and seven.
A birdie at 14 saw her tie Henderson for the lead. She was tied with Thompson atop the board when she arrived at 17, falling back with a bogey before a birdie at the par-five 18th at Las Colinas.
"Ball-striking was I think pretty solid," said the former world number one, whose 13 US LPGA titles include five major championships. "I mis-drove it a couple times and that brought me a little bit of trouble, so I made a couple bogeys," said Park. "There were birdie chances there that were just so close, but just didn't go in."
Thompson, whose four career victories include one major, opened with a birdie at the first. She picked up another shot at the par-five seventh, and gained her first share of the lead with a birdie at the 10th.
The 20-year-old tripped to her lone bogey at the 11th, and also birdied 18 for her share of the lead.
"Overall, it was a very steady round," Thompson said. "I mean, my back nine was a little shaky, missed a few putts here and there, but overall I struck it well off the tee and hit a few good iron shots, so just waiting for those putts to drop."
Webb had the round of the day, nabbing six of her nine birdies on the front nine with two bogeys.
World number one Lydia Ko, who made the halfway cut on the number on Friday, survived a secondary cut that eliminated 19 more players as her one-under 70 left her at even par for the tournament.