Chocolate cake baked from recipe is too delicate/fragile
With a batter like this there are several things that can go wrong. My first guess would be, that the cake would have come out much different had you made the full recipe. When using only a third of the batter you have an increase in surface area compared to the inside of the cake. Not knowing the exact circumstances, I assume your cake dried out a bit too much while baking, especially around the edge. Higher cakes tend to be a bit more moist and also a bit more dense, which makes them more stable.  

Another thing could be, that you didn't let the cake cool enough, before removing the parchment.