I don't think that there are any mandatory requirements (such as a specific degree) for being qualified as a paralegal in France.
Most of the paralegals I've worked with have first gained experience as an assistant in legal departments, and have slowly gained more independance in dealing with some legal matters (the drafting of simple agreements like non-disclosure agreements, transfer of material on a free of charge basis, etc. based on templates), dealing with some trademarks-related matters, and delegation of powers, etc.) but it has indeed taken them many years to get to this position.
However, by looking at the job offers for paralegal positions, the qualifications that are most often requested seem to be:
-a bachelor's degree in law
-a 4/5 years' degree in any field + similar experience
-a degree in law + a degree in legal assistantship
All in all, I think that with your experience in legal administrative position + being bilingual, completed by having a degree in law (either French or English law - French organism CNED offers an online bachelors degree formation in law by the way), you shouldn't have too much trouble finding a position as a paralegal.
I hope that the info provided helps you a little. I'm not a paralegal myself so my knowledge on the matter is a little vague, sorry for that!