Pärson, who has already claimed two bronze medals here at the Games, held her nerve in the fog-hit second leg of the race to finish ahead of Austrian pair Nicole Hosp who was second and Marlies Schild, who took the bronze medal.
Kostelic, who had struggled in the first leg finishing 0.69 seconds behind a determined Pärson, battled in the second slalom but could only finish fourth, missing the bronze medal by just 0.15sec.
Pärson meanwhile was delighted having won a first Olympic crown in what many expect will be a hugely successful career.
After finishing her race 0.29 seconds behind Hosp, the second last skier to leave the gate, the 24-year-old reigning overall World Cup champion threw her arms up in the air in triumph.
She then walked up the slope before running down and carrying out her customary belly flop, much to the delight of her watching father Anders, her coach and a member of the Swedish team.
The victory proved welcome in more ways than one for the Swedes, their last Olympic medal in women’s alpine events going back to 1994 when Pernilla Wiberg won the women’s combined in Lillehammer, Norway.
Kostelic’s failure to medal meanwhile means she will not be able to match the Olympic medals record in alpine events, currently held by Norwegian Kjetil Andre Aamodt, who has eight medals from five editions of the Olympics.
Kostelic, who has won a gold medal here in the combined, has a total of six Olympic medals – three golds and a silver coming from her legendary performances at Salt Lake City in 2002.