Paris for family of 6

Scalajo - Try Hotel Latin Excelsior in the Latin Quarter (near Pantheon, Notre Dame, etc) and book one of their family suites. It's centrally located, clean, friendly, and in a great neighborhood.

https://www.excelsior-paris-hotel.com/

As well as this hotel, they have a family suite that sleeps 6.

https://lecitizenhotel.com/en/

Look into this hotel too.

http://www.hotel-duquesne-eiffel-paris.com/en/

Don’t limit your search to Air B&B. Try VRBO and Home Away as well.

No matter which agency you use for an apartment, make sure it is a legal rental.

As Lola notes, the rental, to be legal, must have a 13 digit registration number. That will be featured on the AirB&B page or the VRBO page. If the number isn't there, you risk not having accommodations when you arrive. Caveat Emptor.

Uhm I"m going to piggyback on this. I have an airbnb reservation for august - the first time I've used Airbnb. I do not see one of these registration numbers and now I am nervous. Should I be? If so, what should I do at this point?

You might also look at apartment rental agencies such as Ville et Village or France Homestyle. Although more expensive we had good experiences with both some years ago.

Difficult to answer about renting illegal apartments. The French government and the city of Paris have put a lot of effort into preserving scarce housing resources for residents. They have also attempted to maintain an apartment owner´s ability to supplement his income by renting his residence short term when he is not there. A complete ban, as other cities have done, would greatly simplify enforcement.

It's a difficult balance and the situation is exasperated by people knowingly renting illegal properties.