Delta is slightly more fuel efficient than new AA. The following data is for consolidated operations (mainline and regionals) and yes, the mainline carrier pays for the fuel used by contract regionals.
For the first 9 months of 2014, DL flew 61.60 ASMs per gallon.
For the first 9 months of 2014, AA flew 61.26 ASMs per gallon.
Advantage Delta, but it's a very small advantage. Both airlines have a wide mix of old fuel hogs and newer, more fuel efficient jets. Both airlines have large fleets of smaller, less efficient RJs and both airlines have large fleets of larger, slightly more efficient 76-seaters.
The more important issue is fuel cost per gallon, where the major airlines vary substantially. For DL, the refinery and successful hedging can lower its fuel costs. For new AA, the absence of hedges (Parker doesn't believe it's a successful long-term strategy worth the huge potential downside) works when fuel costs are declining but could bite new AA in the ass if fuel costs spike again.