I would argue that the story is actually not concerned with presenting a positive or negative view of civil rights struggles. Rather, it draws on material from the history of African-American enslavement in the United States to make a point about a different question that Heinlein was interested in exploring. While the story's treatment of race could certainly be criticized, I find it unlikely that it was intended to be a "racist mirror of the fight for equal rights".
The following passage makes clear that the story itself does not portray a fight for equal rights:
We ask that Jerry's humanity be established as a matter of law. Not for him to vote, nor to hold property, nor to be relieved of special police regulations appropriate to his group—but we do ask that he be adjudged at least as human as that aquarium monstrosity just removed from this court room!
Rather, it focuses on the question "what is a human?"
So I would say Heinlein is using "Old Folks at Home" not as a means of commenting on whether humans of different races are deserving of equal civil rights, but as a means of commenting on whether or how we can recognize humanity in those who are legally treated as property (chattel). I think it's safe to say that he expected his readers to agree that enslaved plantation workers were human beings, so the point seems to be to extrapolate from this basis the conclusion that Jerry in this story is likewise a human being, and deserving of recognition as such. That recognition does not depend on him having equal mental faculties to an average person—any more than the lesser mental capabilities of children mean that they are not human beings—so if this is the purpose of the comparison, then it does not directly hinge on whether or not Heinlein held racist views. (But if he did, that wouldn't be the point he was trying to make here.)
While we can't automatically assume that the end of a story will be a straightforward or unironic statement of the author's point of view, my read is that the following speech by the attorney sums up the theme of the story:
We are exploring the meaning of this strange thing called 'manhood'. We have seen that it is not a matter of shape, nor race, nor planet of birth, nor acuteness of mind. Truly, it cannot be defined, yet it may be experienced. It can reach from heart to heart, from spirit to spirit.
Regardless of the point being made, you aren't the only reader to have been made uncomfortable by this comparison. Here is a blog post I found that discusses this story: you may find it and the comments interesting: "BioSci Fi: "Jerry Was a Man", Robert A. Heinlein, 1947", by Henry T. Greely, October 17, 2012.