Korean War--Who Fought

Feeling nitpicky today so this Times piece upset me.  It's a discussion of Vietnam and Korea relationships, quite good in most resspects.  But this:

Vietnam’s ties with North Korea were strengthened during the Vietnam War, when North Korea dispatched dozens of fighter pilots to combat the Americans. At least 14 North Korean military personnel were killed in action in Vietnam. (About 300,000 South Koreans fought on the American side.)

What's wrong? The last sentence.

Those of us old enough to remember know that technically the United Nations fought against the North Koreans and later the Chinese.  (The Soviets had been boycotting the Security Council so were not around to veto a resolution authorizing UN action against the invaders.)   It was a UN coalition fighting, including British and Turkish troops as I remember it.  (Wikipedia) 

But what really jars is the idea that South Korea fought on the American side.  The war was sold to the U.S. and UN as a fight against the North Korean invaders in which the UN was coming to the aid of South Korea, so we were fighting on South Korea's side.  Since then there have been challenges to that narrative by some historians, but I think the consensus still generally supports the original take on the situation.