Make sure the milk you add to the roux is cold. The sauce will thicken when it comes to the boil again. If you start with cold milk, you have some time to stir the sauce until smooth.
Cook the sauce on a low heat until the flour is cooked. Taste to check.
Add the milk gradually to avoid lumps.
Only add the parsley right before serving or it will turn brown.
This sauce goes well with poached fish.