Solar UV radiation is broadly classified as UV- A,B and C (Decreasing order of wavelengths). Penetrating ability of UV is a not a direct function of its wavelength; lesser the wavelength lesser is the penetrating ability (But higher is the energy: a direct function of wavelength). Penetration ability is dependent on scattering and absorption. Most biomolecules in the skin, including melanin and proteins absorb at around 220-300nm range. Plus, scattering (esp Rayleigh scattering) is inversely proportional to wavelength.
Since UV B has higher energy compared to UV A, the former is more toxic to the cells. But a little UV B also does some good; conversion 7-dehydrocholesterol to Vitamin D. UV A doesn't have sufficient energy to directly cause DNA damage but it can give rise to free radicals and cause DNA damage indirectly.
The paper that you mentioned says that the tissue that they studied has more of UV A mediated lesions. This could be because of higher availability of UV A to the cells. Ozone filters out a major chunk of UV B and C and melanin does little more. If these mechanisms didn't exist then high energy UVs would have burnt us.
To summarize, the inherent order of toxicity is
C > B > A