Place bacon in a large, deep skillet and cook over medium-high heat, turning occasionally, until evenly browned, about 10 minutes. Drain the bacon slices on a paper towel-lined plate.
When bacon is cool, crumble and set aside in a bowl.
Place asparagus spears into a large skillet and pour in about 1 inch of water. Place over medium heat, cover the skillet, and simmer asparagus until bright green and slightly tender, about 5 minutes. Remove asparagus and set aside.
Place cream cheese and chives into the bowl with bacon and stir to evenly combine.
Roll each bread slice with a rolling pin on a work surface until the slices are thin and flat.
Spread cream cheese mixture evenly over each slice of bread.
Center 2 asparagus spears on a bread slice, and roll the slice around the asparagus into a tidy and compact cylinder. Place seam side down onto the prepared baking sheet. Repeat with remaining ingredients.
Brush each roll with melted butter and sprinkle each with about 1 1/2 teaspoon Parmesan cheese.
Bake in the preheated oven until lightly golden brown and hot, 10 to 12 minutes.