A motorcycle center-stand is intended primarily for maintenance work, not for routine parking of a bike. It allows you to get one (or both) of the wheels off the ground to enable chain maintenance or removing one or both of the wheels. By default (on most modern bikes) it is the rear wheel that is lifted. By lifting the rear wheel, you will no longer have any of the “parking brake” effect of putting the bike in gear to help prevent the bike from rolling forward.
The side-stand is the one intended to be used for parking. It is considerably more stable laterally than a center stand. It takes much more effort to high side a bike (push over to the right) from the left side, side-stand than it is to tip it sideways off of the relatively narrow footing of a center-stand. And it is completely stable to the left against the stand’s wide leg, so long as the bike doesn’t roll forward. This is why when you take a ferry over rough water they always have you put the bike on the side stand – NEVER on the center-stand. They also sometimes take precautions to keep the bike from rolling forward (chocks) and high siding (strap bike against the side-stand to the left). With the bike in gear, a side-stand is also more stable in the forward direction of the wheels.
Both stands are devices that only lock into place by the gravity of the bike being “over centered”, i.e. past the highest point of the stand’s deployment arc. In both cases the stand folds to the rear, so that if the bike is pointing downhill (even a small amount) it is more likely that it will want to come off of the stand. If the bike is pointing uphill it is actually less likely that it will want to come off the stand than it is on level ground, and in fact it can be difficult to get a bigger bike off the center stand if the hill is too steep.
So, what does all of this mean? What are the takeaways?
It means that you should never intentionally park a bike with its nose pointing downhill. Even on the side stand where you can put the bike in gear, if left long enough the weight may overcome the resistance of engine compression and eventually creep forward far enough to go past center and your bike will take an asphalt nap. Learn to judge a potential parking spot’s slope and back into any spots that would be nose down. That also makes getting back out of the spot later much easier than trying to paddle the bike uphill backwards.
It means that, especially in soft or irregular parking areas, or when there are high winds, just about any time that you are parking and not intending on doing maintenance (ie like lubing the chain at a gas stop), you should just use the side-stand and leave the bike in gear.
If you push forward on the bike, (like when plugging a tire) it is quite possible to push the bike off of either type of stand. For any maintenance that requires applying force from the rear of the bike, the stand should be “locked” into the forward position, like by strapping it to some forward, immovable part of the bike. The center stand (if you have one) is merely a convenience item. It is always there for your use while on the road. It is not intended to be the end-all of stability without taking the appropriate measures.
The one place that I do routinely use my center stand for longer term parking of my bikes is in my own garage, where the floor is flat, smooth and level concrete, and I have complete control over the surroundings. Doing so allows me to fit more motorcycles into a given space and to get the bikes’ weight up off their tires for longer periods of storage (like winter, ugh!).