I'm afraid this is not correct.
Enable, who manage Wandsworth Common on behalf of Wandsworth Council, say this on their website:
"If your event involves fewer than 50 people you don’t need special permission but you must consider and abide by the following:
* You may not put up a gazebo or any other structure in a park; use the natural shade/shelter provided by trees"
Best to call Enable and ask, if you want to be on the safe side.