The final major of the year is the Evian Championship, played at Evian-les-Bains, France next month.
Park picked up seven shots in her last 12 holes for a 7-under 65 to reel in Jin-Young Ko, who lost the lead at Turnberry when her second shot at No 16 rolled into the burn. Park finished on 12-under 276.
The LPGA Tour is calling Park's achievement a career Grand Slam, although Park hasn't won the Evian Championship since it was given the status of a fifth major in 2013. She did win Evian in 2012.
"This is something I have been dreaming of all my life and all my career," Park said. "To finally reach my goal is just a relief ... I don't know what else to go for."
Park won her seventh major overall, and has captured six of the last 14.
Jin-Young Ko began the final round tied for the lead with Teresa Lu, and made an eagle and two birdies around the turn to go three shots clear after 10 holes.
Park pulled away from a chasing pack that included Lydia Ko and Suzann Pettersen.
As Jin-Young Ko began to tighten, Park made her move. She rolled in an eagle putt from 6m at No 14 to close to within one shot, Ko missed a par putt on No 13 soon after and Park holed a short putt for birdie at No 16 to take the lead for the first time.
Ko's chances of reclaiming the lead virtually ended when she hit her approach shot on the par-4 No 16 right and into a stream.
Ryan Fox continued his fine form on the European Challenge Tour with a 10th placing at the Madeira islands Open in Portugal.
Fox, who won his maiden event on the tour in France last week, finished nine shots back from winner Roope Kakko of Finland.
The New Zealander fired a seven-under 65 yesterday to finish at 13-under and move up 18 places on the final day.
Fox had nine birdies in his final round, six of them coming in a row.
The next stop on the Challenge Tour is the Northern Ireland Open.