It’s on the Vichy France Reference Sheet for sure. On the Reference Sheet under “Vichy Surrender”"
"If Southern France is conquered, Vichy surrenders. All Vichy colonies subsequently join Free France. All Vichy units join Free France. 14.4.5 Internal French Conflict Vichy and Free France are in an internal French conflict. They may attack each other’s units and land zones. A Free France attack on Vichy does not make Vichy Align with Germany.
If Vichy attacks a Free French land zone containing units not from Free France, these non Free French units do not participate in the combat and are moved to the nearest land zone in their possession.
If Vichy attacks Free France in a sea zone containing other Allied naval units, these non Free French units do not participate in the combat but stays in the zone.
The bolded part is what’s really relevant. Now that I look at this, the rest of that doesn’t really have anything to do with Vichy Surrender, so not really sure why it’s under that section on the Reference Sheet haha.
Long story short, retaking Paris doesn’t do anything in terms of Vichy. You need to retake Southern France, which is the Vichy capital.
The Rulebook under 14.4.4 Vichy Surrender also states:
“If Southern France is conquered, Vichy surrenders. All Vichy colonies subsequently join Free France”.
Hope that answers your question!