Upgrades on a codeshare booking?
 

I am a DL Gold member and booked an int'l flight to ICN through DL. What's the best way to book upgrades (miles? medallion complimentary?)

TIA.

Definitely not medallion complimentary. Totally not available. Not even Platinum Medallions can get upgraded using PMU's. And I'm 99% certain that you cannot use mileage to upgrade either. Delta upgrade awards are valid only on Delta-operated flights (I presume this is a DL-coded, KE-operated flight, based on where you've posted the question). And Delta does not have upgrade awards with Korean, which wouldn't be available anyway since you've got a DL-coded ticket.

Sorry to bring bad news, but neither option you suggested works. I'm not aware of any other options that would work, under the circumstances.