During which of Mead's stages of the self does a child begin to "become" a doctor, a parent, a superhero, or a ship captain?

The Play Stage.

Explanation:

George Herbert Mead developed a theory of Self, in which he proposed that how children change their perceptions about themselves.

The theory of Self is divided into three stages, namely, The Preparatory Stage, The Play Stage, and The Game Stage.

The stage in which a child wants to become 'a doctor', a parent, etc is the play stage.

According to Mead's theory of Self, in The Play Stage a child between the age of 2-6 years, starts playing the pretending game. In such games, a child would play game pretending to be a parent, a teacher, a doctor, a scientist, etc. They play the roles as they observe them in their surroundings, either real life or reel life.

So, the correct answer is the Play Stage.