Why does UVA penetrate deeper than UVB, though it's weaker?
Our skin contains quite a lot of water and I think this is answers the question: The absorption of UV light by this water goes drastically up when you shift to shorter wavelengts with a minimum absorption in the UV-A spectrum. 
See this figure (from here):



So water has a very low absorption around 340-350nm (UV-A) and a very high absorption at shorter wavelength (UV-B and UV-C). This also explains why we get a lot more UV-A through the atmosphere than for harder UV-Radiation.