I would like to know if I would have problems getting the VWP in case I travel with my Korean passport.
It depends on which other country you are a citizen of. A few years ago, the VWP was changed such that those who hold a VWP-eligible passport who also have the nationality of either Iraq, Syria, Iran, or Sudan are ineligible to use the VWP:
Under the Act, travelers in the following categories are no longer eligible to travel or be admitted to the United States, without a waiver, under the VWP:
- Nationals of VWP countries who have traveled to or been present in Iraq, Syria, Iran, Sudan, Libya, Somalia, or Yemen at any time on or after March 1, 2011 (with limited exceptions); and
- Nationals of VWP countries who are also nationals of Iraq, Syria, Iran, or Sudan.
Otherwise, holding a passport from a country whose nationals aren't eligible for the VWP (such as Paraguay) does not disqualify you from the VWP.