In a small bowl, dissolve the yeast in 2/3 cup warm water and set aside.
In an extra-large bowl, combine the mashed potatoes with the remaining 2/3 cup water, sugar, softened butter, eggs, cheese and salt.
Add the yeast mixture to the bowl, along with about 2 cups flour, and mix until smooth. Continue adding flour to the bowl until you have a soft, slightly sticky dough. You may not need the full 6 cups of flour.
Cover the dough with a kitchen towel or plastic wrap and allow the dough to rise until double in size. This will take about an hour.
Push dough down, re-cover and allow to rest for about 10 minutes.
Grease a baking pan (with a pat or two of butter or cooking spray) and set aside.
Pull portions of this raised dough, shape it into balls and place them into the prepared baking pans. You should be able to get about 40 potato rolls from this dough.
Cover with a kitchen towel or plastic wrap and allow the rolls to rise for 30 to 45 minutes, or until rolls are not quite double in size.
Preheat oven to 350° F.
Bake rolls for 20 to 25 minutes.
Toward the end of the baking time, prepare the topping by mixing the melted butter, garlic powder, parsley and grated parmesan cheese in a small bowl. Make sure butter is not too hot or it will melt the cheese and make it difficult to spread.
Remove pan from the oven and brush tops of rolls with the melted butter/parmesan mixture.
Return to the oven and bake another 5 minutes, or until rolls are golden brown.