If the LFS had the fish QTed in copper treated tanks and most do, then it is probably safe to assume there is a low level of parasites present in your DT and the stress of transport and acclimation left the tang susceptible to infection. A common problem with tangs.
Your other fish being established and well nourished are able to ward off these parasites for the most part but low level, non lethal, unnoticeable infections continue so the parasites do not completely die off in your system. This is undoubtedly the condition that occurs in the wild or the various parasites would quickly become extinct. IMO low level parasitic infection is present in most of our system but like in the wild are non lethal to otherwise healthy, well nourished, unstressed specimens.
All of this is why IMO for tanks where tangs are an intended addition that they should be introduced first to new tanks that by all logic should be free of parasites due to the length of the cycling period. This also assumes that the tangs were bought at a reputable LFS that runs carefully monitored copper levels in there holding tanks and that copper exposure has not been to lengthy.
After the tang(s) have become established and well fed in your care the addition of other carefully QTed fish can commence. This assuming that no evidence of parasitic infection has been noticed in the previously introduced tangs. Only draw back here is if the tangs are infected you are back to a month of fish less tank to guarantee the die off of those parasites before other additions can be made.
All of this is in the hope that if at a later date parasites are inadvertently introduced to the system and there are umpteen ways they can be. The established tangs will be able to ward off those parasites. Other specimens that are not so naturally prone to infection most certainly will be able to deal with it.
Bottom line is you may have a difficult time introducing any new tang to this system even after a QT period for the tang especially now that the parasite population in your tank has grown due to there ability to lethally infect a compromised specimen.
But then again, that's just me. I could be wrong
Call me crazy but the presence of low level population of parasites is a natural condition, it certainly is in the wild and could actually have some unknown benefits even in the enclosed system.