They connect abstractions and interactors. A parser translates changes broadcast by an interactor before sending them to the abstraction and an unparser translates changes received from an abstraction before sending them to its interactor. E.g: a budget/text editor abstraction/interactor pair. Changes to fields of the budget are unparsed to changes to the text editor and similarly changes to the text editor are parsed into changes to the budget fields.
Parses/Unparsers may be combined with in/out couplers. They may be associated with interactor binders. Example of generic parser/unparser pairs: Consider a a Suite-like structure editor that traverses the fields of an arbitrary object and creates a tree of text/slider interactors based on the types of the leaf level nodes. A generic parser/unparser can be written for such an editor.