Melt butter in saucepan over medium heat; cook and stir onion until lightly browned, about 10 minutes. Add potatoes, water, dill weed, salt, and pepper to onion; bring to a boil. Reduce heat and simmer until potatoes are tender and water is almost all evaporated, at least 30 minutes.
Stir milk and bacon bits into potato mixture; bring to a boil. Remove from heat and stir parsley into soup.