I'd use some 10%bleach and soak it. Rinse it off, sand it and use some sort of wood treatment after it dries in the sun. Perhaps some deck stain, or deck sealer.
You need to keep the wood from holding / soaking up water.
Those look to be lichens, not mold. There are lots of Google hits on the subject of their removal and subsequent prevention.
Some sort of chemical application is probably needed if you cannot use other measures to keep it away (avoid damp, shady conditions).
Wet and Forget has a chemical ingredient ( dimethylbenzylammonium chloride) similar to what we use in greenhouses designed and labeled to reduce algae growth. Follow directions for safety - on second thought, you might want to check for child friendliness.