The documentation on Adobe Character Animator Help | Control puppets using behaviors says "Parallax Strength controls parallax movement of the eyes, nose, and mouth" but I have never got the mouth to move.
So either the docs need fixing or the software.
Mouth strength is not useful - on Chloe it distorts the mouth badly.
I worked out a hack - you restructure the artwork to make the Mouth a child of Nose, then when the nose is moved, the mouth moves as well with the nose.