Q:

Is Vitamin D production related to skin pigmentation?

Alice Roberts published an article in the Guardian newspaper, titled shining a fresh light on vitamin D
She makes the point that the prevailing theory is that skin lightened in regions away from the equator so that skin was sensitive enough to make vitamin D. But she also cited a paper which appeared to show that vitamin D production isn't dependent upon skin pigmentation.
Is Vitamin D production related to skin pigmentation?

A:

Yes.
Prevalence of Vitamin D Deficiency and Its Associations with Skin Color in Pregnant Women in the First Trimester in a Sample from Switzerland concludes "In the most parsimonious model, women with dark skin color were statistically significantly more often vitamin D deficient compared to women with light skin color..."
Similarly, Colour Counts: Sunlight and Skin Type as Drivers of Vitamin D Deficiency at UK Latitudes puts forward that:

The melanin pigment absorbs ultraviolet radiation (UVR) and protects
  underlying skin from damage caused by UVR. It also reduces the UVR
  available for vitamin D synthesis in the skin. It has been shown that
  the white-skinned population of the UK are able to meet their vitamin
  D needs with short, daily lunchtime exposures to sunlight.

and concludes

Results show that to meet vitamin D requirements, [Fitzpatrick] skin type V [(brown)]
  individuals in the UK need ~25 min daily sunlight at lunchtime, from
  March to September. This makes several assumptions, including that
  forearms and lower legs are exposed June–August; only exposing hands
  and face at this time is inadequate. For practical and cultural
  reasons, enhanced oral intake of vitamin D should be considered for
  this population.

