need parallax effect for mouth too

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.

If you increase Mouth Strength (and have the Mouth group as independent) it should do some parallax. Mouth Strength will also scale the mouth, though.