Do I have to manually add (animator controller) references to new states and parameters in an animation script every time I add new states and parameters to an animator controller?

In the unity animation tutorial, they say that animator controllers have state machines that switch between states based on a given parameter provided by an animation script. That said, do I have to add a piece of code to an animation script every time I add a new parameter or state in the animator controller or is there a way for Unity to do it automatically?

As far as I know, Unity does not do that automatically. Exactly as you said, you have to add a code reference every time you add/modify something in the animator controller.