I say this from experience. I am a black person who is not particularly dark skinned, and have only had the slightest sunburn once in my life, after I had been working in a field for 16 hours straight in full sun under clear sky detasseling corn. When I got back to camp, the back of my neck seemed slightly sensitive, which was unusual. It was fine the next morning, and never peeled at all. I did the same for 2 weeks, and this was the only time I felt this. This was the worst sunburn (if it was one) that I've gotten in 50 years of life. I've spent plenty of time outside, and plenty of time inside being a code monkey and never going outside. I've never worn sunscreen.
I've never met a black person as brown as a paper bag who ever got a sunburn, at least that I know of. My dad is pretty pale, I think I would have heard about it.