In trying to justify its own nuclear program, North Korea is trotting out a familiar, but false canard: we need nuclear weapons to "protect ourselves" from U.S. nukes in the south.
I hate to disappoint the boys in Pyongyang, but that dog won't hunt. In the early 1990s, I was part of a team that handled the removal of U.S. nuclear weapons from South Korea, and I've seen no evidence that they subsequently returned to the peninsula. Heck, the Washington Times (and Defense Secretary Don Rumfeld) confirmed the pull-out more than a decade later. Moreover, I'd be greatly surprised if North Korea was unaware of that removal operation, given their extensive spy network in the ROK, and their periodic access to Russian and Chinese intelligence information.
At the time, the removal was justified by a variety of factors, including reduced maintenance costs, our ability to quickly reintroduce the weapons if required, and the presence of nuclear weapons on "other" platforms (read: naval vessels) that operate in the area. That policy likely remains in effect to this day.
The U.S. probably has storage facilities for nuclear weapons in South Korea (as it does in many countries) but there is no compelling evidence that those sites are currently housing nukes. And, you'll note that Pyongyang offers no specific information to support its claims. Nice try, boys, but no cigar. If we ever need to turn the DPRK into a giant glass parking lot, we've got plenty of nukes to do job, and many are "already in the neighborhood," so to speak.