Place milk, parsley stems, carrot, and onion in a medium saucepan over medium-high heat. Bring to a simmer and season with salt and pepper; let simmer for 4 to 5 minutes.
Remove saucepan from heat and strain; discard solids. Return milk to saucepan and bring to a boil over medium-high heat. Whisk in roux until mixture is thick enough to coat the back of a spoon; season with salt and pepper. Add chopped parsley and reduce heat to very low; simmer for 4 to 5 minutes.