I agree that is the common answer but it is not a very scientific one. Given the structure of wood, water does not flow through it in any normal sense, it moves via capillary action and vapour transfer neither of which are much affected by gravity at the moisture content typically encountered in a propeller. The idea that all the water in the prop is going to run down through it to one end is just a bit silly.
A better question might be to think about the possible effects on water entry to the propeller. The idea being that we would want to minimise this to keep the internal moisture content as stable as possible.
If the aeroplane is in a hangar, it is not getting wet so it really does not matter, from a moisture point of view, how it is aligned. You may as well leave it where it stops.
If it is outside and going to get rain or dew on it, which way should it go? If you put it horizontal then the leading edges are going to be more exposed to water either from rain directly on the upper edge or from drops of condensation collecting on the lower edge. And where is the protective finish likely to be damaged? If you have a metal protective strip then you definitely don´t want a leading edge down as water will get in under the protective metal at any break in the seal between metal and wood, the area will remain damp and the wood will be damaged. Horizontal is not good.
If you put it vertical then moisture does not stand on the leading edges and any that gets under protective strips has a chance to drain. Proper metal protective edges have drain holes to facilitate this. However, as a previous post mentioned, water can run down to the hub and possibly get under the mounting flanges, become trapped and cause swelling and other problems. Vertical is not good.
All is not lost however. If you put it on a diagonal then you get the best of both worlds, the edges drain and contours around the blade root will deflect most moisture running down the blade around the hub area so the mounting flanges stay dry.
There are plenty of pictures of rows of tigers and the like, all with wooden props parked at an angle, taken in the days when they knew a thing or two about operating wooden props.
As others have mentioned, you also get the other benefits. The propeller is less exposed to hangar rash, being walked into, being used as a perch to encourage sparrows to nest in the engine or being used as a towbar to drag the attached aeroplane around. This last is an all too common practice which has far more potential to damage a wooden propeller than has any effect of how it is parked.
Finally, a few words on respect for propellers from someone who spent several years operating with Gypsy Majors including hand starting the things. The big lesson learned from that was to never trust a propeller, there was an even chance that it had homicidal tendencies. It makes me cringe when Biggles arrives shuts down the engine, hops out, grabs hold of the prop, pulls it through a couple of compressions to get it to whatever position he has been taught then drags it off to his parking spot with a hand clamped around, usually one blade. It happens all over the place every weekend. For any engine with impulse magnetos this is just Russian roulette with a blunt axe. Even with electronic ignition engines there is a real risk. They may be hard to start but it is only a matter of time before the thing gets a bit of glowing carbon in it, there is no mixture cutoff so there is plenty of fuel there and it pops over a few compressions. That´s all it takes to cause smashed fingers and serious cuts, not to mention incidental prop damage.
I´ve seen it happen, it has happened to me and the only reason I still have all my fingers is because I assumed it was going to and used proper technique to get my hand out of the way before the next blade came around.
Please, don´t treat your propeller like a handle, treat it like the dangerous tool that it is.