Clark moved to Rugby Park from Queen of the South last summer, along with manager Allan Johnston when he replaced Kenny Shiels as boss.
The 57-year-old signed a two-year contract but he will now depart the club after one season in which Killie finished ninth in the Scottish Premiership after ending the campaign with crucial back-to-back victories over St Mirren and Hibernian to avoid the relegation play-off.
A statement from Kilmarnock read: "The club would like to thank Sandy for his diligence and professionalism over season 2013-14.
"In testing circumstances, his wealth of experience was an important factor in the club pulling clear of the Premiership relegation play-off as the battle to avoid 11th place came to an exciting conclusion.
"First-team manager Allan Johnstoný wishes to record his disappointment that Sandy has decided to leave the club and extends his sincere thanks to Sandy for his dedication in making their partnership so successful at Queen of the South, winning two trophies and helping Allan to achieve the 2012-13 'Manager of the Year' award.
"Allan acknowledges that the step up to Premiership management in 2013 was a challenge and pays tribute to Sandy for his role in helping him through that transition.
"He wishes Sandy every success in the future, as do the club's directors, staff and supporters."