You'll be able to take a stopover, but it will cost you.
If you're leaving in late december (after 16Dec) the cheapeast TW or AA fare JFK/CAI is $1400. However, it doesn't allow for stopovers. The cheapest TW or AA fare that I can find that allows for stopovers in late december is a $1878 fare, which allows for a single stop-over for an additional cost of $100.
Roughly, this means you can go JFK-CAI and then CAI-JFK for about $1410USD with taxes, or JFK-PAR, PAR-CAI, CAI-JFK for about $1988USD.
A much cheaper alternative would be to buy one set of tickets JFK-CDG-JFK, which in late december price out under $400, and then another set of tickets CDG-CAI-CDG, which for late december are under $500 for major carriers. This gets you the entire itinerary for under $1,000. It also means you can, should you chose, stop-over in Paris (or possibly another European city) in both directions...
The downside is that you won't end up with AAdvantage or Aviators points for the CDG/CAI legs because the BA fare for that market is $700, more than $200 over what it would cost on AF/LH/SR. However, even if you did it on BA to get AAdvantage miles, the total cost would still be under $1300/person versus $1988/person using the JFK-CAI (return) fare.