AZ is the code for Alitalia, much like American Airlines uses AA or Delta uses DL. Alitalia is an airline operating out of Rome Fiumicino. Alitalia is in the same airline alliance as Delta, Air France (AF) and KLM (KL), amongst others. If you stick around FlyerTalk you will notice we sometimes descend into incomprehensible acronyms.

In principle you can use your Delta miles to get an award ticket on AZ (or anyone else in the alliance), but given AZ's inclination to void even premium revenue tickets apparently at random I imagine it would be a challenging exercise even for a seasoned professional. But perhaps they have wide open award space?

If you and your wife are not adverse to taking connections you might find award availability is higher if you are happy to fly somewhere nearby before heading onto Paris.

I would also suggest you keep a close eye on revenue prices, although obviously that isn't as fun as getting a ticket for free.