This social phenomenon is present not only among other Asians, but also blacks, whites, Hispanics, Arabs, and others. Is this a form of racism? Of course the answer is, it depends.
As much as I care about older Koreans (especially my family), there is a certain amount of racism in their attitudes about dating and marriage. Many believe the purity of Korean heritage is special and that it would be contaminated by having children with non-Koreans. They may not think that Koreans are better than other ethnic groups, but certainly better for their children to reproduce with. That is at least a bit racist.
The good news is that this mentality has changed significantly in America and is continuing to change. Korean parents are becoming much more accepting of exogamy and their primary concerns are about whether the family will be happy with having a non-Korean and still trying to maintain their family traditions. I am not entirely sure how this change occurred but I would credit it mostly to being in a more ethnically diverse nation. Perhaps the racism they experienced as immigrant minorities in America made them realize that treating and judging people purely based on what they look like is unjust so they reflected on how they have been prejudiced about non-Koreans. Maybe they befriended non-Koreans and realized some of them are good people too.
However, it is a bit more difficult to consider the Korean children as having racist sentiments for only dating Koreans. Yes, some may definitely have the same views their parents do about interracial dating but more often than not, they feel they have an obligation to their parents to not date people of other ethnicities. They will almost certainly feel attraction to people who are not Korean but will think that it would be burdensome and to some degree disrespectful to put that pressure on their families.
Fortunately, this is changing as well. Statistically, Asian-Americans out-marry at higher rates than any other racial demographic in America. This is likely due to the fact that Asians are less than 6% of the US population. We (not just Asians) are genetically programmed to effectively make more copies of our genes and that requires a certain amount of environmental adaptability. The parents may be upset at first but what are they going to do? Nag? They will do that anyways. Your kids will be their grandkids so that's a quarter of their genes floating around in each of these new bodies.
We have made great progress in mitigating racism in America. The American government no longer kills Native Americans, sanctions slavery and segregation of African-Americans, and forces Japanese-Americans into internment camps. We now have black, Hispanic, and Asian American government officials and a biracial President. Our iconic celebrities include multiracial Americans like Whitney Houston, Bruce Lee, Jimi Hendrix, Johnny Depp, Oprah Winfrey, Muhammad Ali, and Beyonce Knowles. Even if you are a multi-billionaire, you will be called out and condemned for racist statements (e.g. Donald Sterling). Eliminating negative prejudices about interracial dating will help us form a more tolerant, happy, and peaceful United States.