Jung Yeon Lee, who matched the Dell Urich course record with a 60 in the opening round, managed only an even-par 70 on Friday to join Laura Davies in a tie for second at 10-under-par 130. Seol-An Jeon was two shots further back at 8-under-par 132.
Stupples was three shots behind Lee to start the second round and picked up her first birdie at the par-5 13th, her fourth hole of the day.
At the par-4 14th, Stupples landed a 9-iron inside 18 feet and drained the birdie putt to move to minus-9.
Stupples added a birdie at the par-5 sixth and closed her bogey-free round with a birdie at the par-4 eighth after her approach stopped within 20 feet of the cup.
'It was a day for being patient and biding my time,' said Stupples. 'The yardages for me today always seemed to be in between, and sometimes committing to the shots was a little tougher today.'
Lee also played the back side first and jumped out of the gate with a birdie at the 10th.
The Korean struggled throughout the day, however, and tallied back-to-back bogeys starting at the 17th to fall back to 9 under.
Lee recovered with a birdie at the third and pulled ahead with a birdie at the sixth.
The 25-year-old found trouble again coming home and bogeyed the ninth to finish one shot back.
'It was a struggle,' said Lee. 'It was real tough today.'
Davies started on the 10th as well and collected a lone birdie over her first nine holes.
She then caught fire on the front side with four birdies for her second straight 65.
'The front nine is much easier than the back,' said Davies. 'I did get a few birdies to finish off the day, but that's what you're going to have to do the rest of the week.'
Aree Song fired a 64 to join Grace Park, Heather Bowie and Stacy Prammanasudh in a tie for fifth at 5-under-par 135.
Karrie Webb, Angela Stanford, Vicki Goetze-Ackerman and Jean Bartholomew followed at 4-under-par 136.
The 36-hole cut fell at 2-over-par 142 with 85 players qualifying for the weekend. Last year's champion Wendy Doolan was not as fortunate, however, and finished at 4-over-par 144.