It would help if we knew what sense you were using the name "Southwark" as it is both a place in its own right and a local government area.

If you mean the area north of Elephant then you are going to be in the congestion charge zone and normal car use will be stupidly expensive even if you have parking available.

If on the other hand you mean somewhere in the south of the borough like Dulwich then it might be a different matter depending on the exact public transport options for your journey.