Preheat oven to 400 degrees F. With a damp paper towel, wipe down the mushrooms. Pull out the stems and finely chop them. Set aside. Place mushroom caps on prepared baking sheet.
Preheat medium skillet on medium-high heat and swirl olive oil to coat. Add onion, garlic, mushroom stems and a pinch of salt. Saute for 5 minutes, stirring occasionally.
Transfer to a medium bowl along with parsley, 1/2 cup Parmesan cheese, mozzarella cheese, yogurt, 1/4 teaspoon salt and pepper. Stir with spatula to combine.
Using a teaspoon, fill the mushroom caps with stuffing. Place a bit of remaining 1/4 cup Parmesan cheese on top of each mushroom and sprinkle all of them with more salt. Bake for 20 minutes.
Serve hot or cold. I like them both ways.
Notes
Store: Refrigerate in an airtight container for up to 2 days. Do not freeze.
Make ahead: Can be prepared up to 48 hours in advance. Refrigerate mushroom caps and stuffing separately, then fill and bake just before serving. Or you can refrigerate assembled stuffed mushrooms for up to 12 hours and bake before serving. I recommend to add a few minutes to baking time because mushrooms will be chilled.