World number one Park finished on three-under-par after a fast start as she bids to become the first player of either gender to win four majors in a calendar year.
The South Korean was six-under through 10 holes but dropped shots at the 13th, 16th, which she double bogeyed, and 17th before picking up a birdie at the 18th.
Pressel hit seven birdies and one bogey in her opening round at the Old Course in Scotland.
Eight players finished the day at four under, including Scotland's former British Open winner Catriona Matthew, English pair Liz Young and Georgia Hall, and former U.S. Open champion Paula Creamer of the United States.
The 25-year-old Park won the first three majors of 2013 - the Kraft Nabisco title, LPGA Championship and U.S. Open.
Officially, victory at St Andrews this weekend would not constitute a grand slam, since the Evian Masters in September has this year been granted the status of the fifth major.
But securing a fourth successive major in a calendar year would be heralded as an unprecedented achievement.
Park has already matched the feat of Babe Zaharias in 1950 of winning the first three majors of the year.
(Writing by Stephen Wood in London, editing by Ken Ferris)
