I´ve searched a lot here and did not find a solution this: I've made a character with a lot of parallax (800), but when i change the mouth set, mouth parallax is gone. I´ve found the answer while writing here, but decided to post anyway to help others that might encounter the same problem.
First attempt: I've tried both happy and sad sets (swap set between them) parented to the HEAD layer. Mouths do change according to the sound, and parallax works on the default mouth group, but it loses parallax when i trigger the "sad group". In this case i tried to force a parallax using face behaviour but it did nothing.
Second attempt: I've made a "Mouth Group" layer parented to the HEAD layer and happy and sad sets (swap set between them) parented to the "Mouth Group" set. Now parallax works for both of them, but the SAD group is stuck on the NEUTRAL mouth. I've tried tweaking the tags, no success; i also tried adding lipsync behaviour on the "Sad mouth" layer, but it lists "Mouth" null and does not change its visemes.
Third attempt: My mouth group is parented to the HEAD (as usual), but i made a subset inside each viseme with happy/sad mouths. I made a trigger for each of the visemes that uses the same key for everyone and it finally worked!
Very interesting! I have had all sorts of problems with parallex on mouths and basically gave up. Best luck I had was I put the Mouth with expressions underneath the Nose group. The mouth seemed to move with the nose then. But it was hacky and I did not like it. Thanks for sharing another solution, that while a pain to create so many swap sets, is more logical in puppet structure aftewards!