Nevin Galmarini of Switzerland had a half-second lead after the first of two runs in the finals but made an early mistake in the second round and quickly fell behind.
Wild became a Russian citizen after his marriage in 2011 to snowboarder Alena Zavarzina, who moments before Wild won his race had captured a bronze medal in the women's event.
The bronze medal in the men's parallel giant slalom race went to Zan Kosir of Slovenia.
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