They can fly for themselves or another code which is not a code share with DAL.

But for the DAL flights, Delta does the scheduling exclusively.

It gets funny during IROPS. When Delta got so busy with their own mess the DCI carriers would implode. Comair managed to have their whole airline time out a couple of times as did ASA. During critical IROPS crews were known to get "lost" for as long as two weeks. Of course this happened most frequently in the middle of winter when the DCI crew was someplace nice, like Mexico, with limited, or no, communications ability. You got to enjoy conversations like "Where are you?" "Do you have an airplane?" "Why didn't you call us for two weeks?" (eh' you sent out a memo not to call in during IROPS, we were just following Company guidance."

Of course then they would deadhead you home (why?) and back in the good old days the next crew would find major components of the airplane had been stolen while it sat on the ramp. There was a ACA J41 which got stripped so badly it never flew again. People strongly suspected an on field shop had used the airplane for "spares."