The real problem for Parker will be when large institutional investors start calling for the CEO's head. At that point the BOD will be forced to act. So far at least publicly there's been no demand for his ouster.
As far as why many here continue to fly AA? If you live in a hub and have time sensitive travel do you want to be constantly connecting? So for example where I live MIA. Unless you're flying to a DL or one of the few nonstops to non DL hubs there's a connection at ATL. United is very weak at MIA and has no Southeastern hub. So would I and others like me always want to add on 2-4 hours per flight of connection time?