Of course, both parties say that they are in favor of civil rights today. But they define these things very differently. One way to look at this issue is to note that almost all African Americans (more than 90% in many elections) vote for Democratic candidates rather than Republican ones.
Republicans today believe that African Americans have won all of the civil rights that are needed. They believe that discrimination is not something that is very relevant to the lives of African Americans today. Therefore, they oppose programs such as affirmative action for African Americans. They also oppose the use of federal government powers to try to protect black voting power in the various states.
Democrats tend to believe that African Americans continue to need help to remedy the effects of past (and possibly current) discrimination. They believe in affirmative action. They believe in social welfare programs that will, they argue, help African Americans.
In general, Democrats believe in government action taken specifically to help African Americans. Republicans generally do not believe that such actions will be of benefit to African Americans.