I think this is a very reasonable and interesting question.
Dr Who IS piss poor at including characters who are not white. unless you count mickey, martha is the first black companion. she is also a very weak character. torchwood has toshiko, that is IT unless you count iantos dead girlfriend.
rosemary is also correct that the family were depicted as slaves, while jack was tied up a la prometheus. fine that theres a backstory there but the fact is that they were explicitly shown to be not just slaves but triumphant over the "Master". If you layer this onto black american slave society, esp with the use of the word Master and the doctor as messiah blah blah then I think, yes, you find parallels.
I think in context of the whole world being enslaved-no he prob isn't racist. but i do think it possible that the writers were drawing, consciously or not, on the culture generated by slavery (eg the word of mouth stories, the power of speech, the redemption of a single person who then redemed the rest etc). this isn't racist, of course. its not racist to honour a culture.
i find it rather disturbing that this has just turned into a troll rant. If you feel uncomfortable with the question-why, exactly?