You have been given some good help.
Looking and booking as one ways will be better.
Do not only look for Paris. Try other Eu cities. Trains between Eu cities are easy & frequent.
If there is availability just take it now. Others will also be looking for USA Eu awards. And AAdvantage flyers will be wanting to avoid BA.

You cannot be too fussy with the hard-soft product. Flying business class is better than coach. The scheduled departure, layover time & arrival times can be a consideration if you have options.
Do not wait to make a decision: just do it. (Awards can be cancelled)

I've never flown AY but my impression from reviews and reports is that no Oneworld option to Europe has an outstanding business class (I've only flown AA A330). Iberia's hard product may be the best while its soft product may be the worst. Yet I wouldn't pay the modest fuel surcharges to fly IB if there's a no fuel surcharge option. My impression is that AA lie flat > AY/AB > AA angle flat, and United similar to AA. In the end, if I were in your shoes I'd use some of that massive pile of AA miles on AY instead of burning UA miles on United metal. UA metal is nothing exceptional and UA miles can be used for much better.

The IB A340s have fully lie-flat seats. The profit surcharge (nothing to do with fuel) is less onerous than on BA. Must call in, cannot see online on aa.com

You haven't bothered to tell us your origin. "US" is pretty large and difficult to deal with. But "LAX" or "Los Angeles area" could include using Air Tahiti Nui nonstop LAX-CDG using AAdvantage miles.

EWR or JFK could include BA subsidiary OpenSkies.

Neither, afaik, will show on aa.com searching for awards. See http://www.flyertalk.com/forum/ameri...ssistance.html, and read the Wikipost at the top of the page for AAdvantage MileSAAver award information.

Made it pretty clear in his first post.

The OP lists the poster's potential US origination airports.

OpenSkies flights definitely show up on aa.com award searches.

I'd take Open Skies even with the BA award charges. You could fly Bizbed on the overnight and premium economy to reduce the charges somewhat. If for 5-6 days I'd avoid a connection. The fuel charges may well be worth compared to the cost of the trip in any case.

You can easily take the train to Paris from BRU or AMS, in addition to FRA (and other airports in Germany such as CGN) and London.

So, if you are trying to get an award ticket to Europe, these are the steps:
1)find transpacific segment. As noted, BA has great availability, just high surcharges. If you are trying to find availability on other carriers try using their bases (for Air Berlin, Dusseldorf, Berlin) and typing in the exact flight route (JFK to DUS for example). Also remember the old US Air flights and type those in (PHL to lots of Europe) as they seem to have better availability than AA flights
2)once you get across the Pond you can basically find your way anywhere. In Europe there are plenty of ways to get places, including BA, AY, AB. AA has very flexible routing rules. In the US I could even find my way across the US from East to West coast. If you type in a search on AA's website (or BA) it won't show you, because it has too many connections, but I found JFK to LAX when there is no availability by using 3 stops. You get 24 hours to connect so have plenty of ways. This is where it takes a lot of legwork, as you have to just keep trying combinations.
3)FYI BA surcharges aren't as high as they once were. There are pretty high surcharges even for flying non-BA, and intra Europe sometimes AA charges $60 or so just for flying Finnair. So actually price it through on AA.com to see what the difference is (and how much extra work you are willing to put in to avoid $100)