In a bowl, combine shiitakes and 1 cup hot water; let stand until shiitakes are soft, 7 to 10 minutes.
As shiitakes soak, peel onion, shallot, and garlic. Finely chop in a food processor or with a knife.
In a 10- to 12-inch frying pan over medium-high heat, frequently stir onion, shallot, garlic, olive oil, and paprika until vegetables begin to brown, 5 to 7 minutes.
Meanwhile, trim off and discard discolored stem ends of common mushrooms. Finely chop mushrooms in a food processor or with a knife.
When shiitakes are soft, squeeze in soaking water to release grit, lift out, and squeeze dry; reserve water. Trim off and discard stems; finely chop shiitakes in a food processor or with a knife.
Add common mushrooms, shiitakes, and thyme to onion mixture. Slowly pour reserved soaking water into pan, leaving sediment behind. Stir often over high heat until mushrooms begin to brown, 8 to 10 minutes. Remove from heat and let cool, about 10 minutes.
Add yogurt, salt, and pepper to taste. Spoon into a bowl and sprinkle with parsley. Serve warm or cool. Spread on crackers to eat.