In nonstick skillet, heat oil over medium heat; add onions and cook, stirring occasionally, until softened, about 10 minutes. Reduce heat to medium-low and cook, stirring often, until evenly golden, about 30 minutes.
Meanwhile, in food processor or using knife, finely chop mushrooms. Add to pan along with garlic, parsley, salt and pepper. Increase heat to medium-high; sauté until no liquid remains, 8 to 10 minutes. Scrape into bowl; let cool.
Stir in sour cream, mayonnaise and half of the chives; transfer to serving bowl. (Make-ahead: Cover and refrigerate for up to 3 days.) Sprinkle with remaining chives.