To expand a bit;
Clouds are not opaque to visible light either. Ever notice how on a cloudy day, you can still see? The amount of light in the visible range that gets through the clouds is not zero. The UV radiation that does the famage to your skin is close to the frequencies of visible light, so the effect of clouds on these two forms of radiation is similar. Not much UV gets through, but some.
That's why the song "My Girl" can truthfully say, "I got sunshine, on a cloudy day..."