Heat a tablespoon of butter and the olive oil in a large pot or Dutch oven over medium heat until the butter melts.
To the pot, add the chopped onion, celery, and carrots. Stir occasionally and cook until they become soft, which takes about 4 to 6 minutes.
Toss in the chopped parsley and stir for about 10 seconds until it becomes fragrant.
Mix in the chopped and peeled potatoes.
Pour in 2 cups of water and 4 cups of chicken broth. Turn up the heat until it's simmering actively.
Reduce the heat to maintain a gentle simmer, and let it cook until the potatoes are very soft approximately 15 minutes.
Carefully blend the soup using an immersion blender or blend it in batches using a regular blender until it's smooth.
Melt the remaining 2 tablespoons of butter in a separate saucepan to make a roux.
Add 2 tablespoons of all-purpose flour to the melted butter. Cook over low heat while whisking constantly until the flour loses its raw smell, which should take about 5 minutes.
Warm up 1 cup of half and half in the microwave until it's hot, but be careful not to boil it. Slowly add a bit of the hot half and half into the hot roux while whisking continuously until the mixture is smooth.
Combine the roux mixture with the remaining hot half and half, stirring to combine.
Gently stir this mixture into the simmering potato soup.
Add the crumbled bacon bits, and season the soup with 1/2 teaspoon of salt and freshly ground black pepper to taste.
Let the soup simmer for an additional 10 to 20 minutes to allow the flour to soften and absorb the liquid. This step is important; if you don't simmer long enough, the roux might stay grainy.
Once the soup is well-mixed and heated through, it's ready to be served hot. You can garnish it with chopped parsley or shredded cheese.
Enjoy your homemade Eat N Park Potato Soup!