The reason is that although light does pass through clouds, it arrives much weaker. Clouds aren't transparent - they reflect, absorb, and scatter a good portion of solar radiation before it reaches the panel. Imagine it's like looking at the sun through a curtain: you can see it's there, but the amount of light that gets through is a fraction of what you'd get without the curtain. Solar panels need photons with enough energy, and clouds dramatically reduce that flow.
Also, a panel's efficiency depends on the intensity of direct radiation. On a clear day they receive direct light and also reflected light from the surroundings. With dense clouds, most of that radiation gets blocked, and what does arrive is scattered. Depending on the type and thickness of the clouds, panels can be producing between 10 and 50 percent of their normal capacity. It's not that they stop working completely, but that's why your dad probably noticed such a big drop when the sky gets cloudy.
The good news is that even on cloudy days the panels generate some energy, so it's never a total loss. In the long run, it adds up.