When is the cheapest time to fly to France from the US?

Take a red-eye flight. Those tend to be cheapest. If you're looking at the best time of year, avoid holidays. It is often the case that February and early March (before Easter travel gets under way) are cheap. Different airlines have different policies, so do a thorough search.

There is another problem. You might encounter problems with the French school holidays where you return flights are going to be booked. So check on those days. That is the low tourist season to good luck with flights. The flights start going up in March. Late March to Early September the flights become more expensive. But they are becoming more expensive anyway.