When I was in college, FW de Klerk was invited as a speaker on campus. He worked in government during the time Nelson Mandela was active in what eventually ended Apartheid in South Africa. I learned from him that at first he and many white people in government sincerely believed it was best to keep black and white people apart and segregated. The motto was “Separate but equal.” The only problem with that notion, however well intended, de Klerk noted, was that it simply didn't work. The inevitable conclusion drawn from countless government meetings and public town halls was that people should not be segregated or treated any differently than others based on race. Limiting your children's contact or level of intimacy with people of other races is treating those people differently.
As for what your parents taught you, I have no doubt it came from a place of good intentions, quite possibly similar to de Klerk's initial notion of separate but equal. However, I bet they would be proud of you for taking the next step in that thinking, which includes the realization that it really doesn't work, and have some thoughtful conversations with your children about what treating people equally really looks like in your situation.