The school situation isn't easy in Paris. If you choose an international (Anglophone) school you have the choice between International School of Paris (ISP), American School of Paris (ASP) and British School of Paris (BSP) and you need to live in the vicinity of the school or on a school bus route to make it work. Costs are eye-watering for an education in a non-selective environment. Marymount (a private American Catholic school) is wildly popular with some parents but only goes up to 8th grade/Year 9. None of the other fully Anglophone schools should be touched with a bargepole.
The highest status school for foreign/dual national families is Ecole Jeannine Manuel. This school is wildly oversubscribed, requires an extremely high standard of bilingualism (plus other languages) and is a fiercely competitive environment that does not suit every family whose children are admitted.
There are many other bilingual, all of which have "issues". The longest standing is Ecole Internationale Bilingue (EIB) Monceau. You need to go to this school with your eyes open, especially since it was taken over by Globeducate, which is an international private equity backed schools group. Ermitage, at Maisons-Laffitte, was a for a long time a complete no go but it has new (again, private equity) shareholders and a new head and is worth investigating. Of the new independent bilingual primary schools, Concordia is the most promising by far.
Many foreign parents in Paris prefer French Catholic (sous contract) schools to the bilingual and international schools. These are academically and socially sound and since they are cheap as chips (even the high status ones, like Franklin or Stanislas) parents retain lots of budget flexibility to top up their children's education. As a general rule in Paris you should expect to buy extra curricular activities and tutoring on the open market (at home tutoring, including music lessons, is 50% tax deductible) - school is quite a bare bones experience.
The French bac is in a poor way and your children should do the IB if they wish to study outside France.