Parnevik, a European Ryder Cup member who now plays most of his golf in the United States, used the fast finish for a 4-under-par 66 and reached the halfway point of the world's oldest golf tournament at 134, 6 under par.
With well over half the 156-man field still out in brisk winds, Greg Turner of New Zealand held second at 136.
Turner, the first-round leader, had to rally to salvage a second-round 71.
Turner had the lead alone through the first seven holes, then ran into difficulty. He was bunkered in two on the eighth, just got out, chipped on and two-putted for a double bogey, then bogeyed the ninth after missing the green.
While he was struggling in the early going, however, defending title-holder Greg Norman and English ace Nick Faldo each mounted rallies: Faldo coming back to respectability and Norman to a place among the leaders.
"It took me 17 holes to undo what I did yesterday, and I made one for myself on the last," Faldo said after a 66 gave him a 141.
After playing the wrong ball and taking a fat 8 on the way to an opening round 75, Faldo was in danger of missing the cut for the first time in 19 British Open appearances when he began the second round. His birdie-4 on the 17th put him 3 under par for the day -- wiping out the triple bogey from the day before -- and he finished it up with a 12-foot birdie putt on the 18th.
Norman, who struggled to a 71 in the first round, came back with a 67 and completed 36 holes over Turnberry's links in 138.
Veteran Fuzzy Zoeller dominated the short holes in a round of 66 that left him three shots off the early pace at 137.
"It's an entirely different golf course out there today," Zoeller said. "The wind has turned around. You have to hit good shots, but the golf course is there for the taking."
He did most of his taking on the par-3 holes, making birdies on all four of them. It set up an unusual card for the veteran, a score of 66 and a scorecard that showed no 3's.
Mark Brooks, on the other hand, had nine 3's in a bogey-free round of 64 that put him in a tie at 138 with Norman, Davis Love III, U.S. Open winner Ernie Els of South Africa and Vijay Singh of Fiji.
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