Be warned. The rules for parking at Strand on the Green have changed. Formerly there were no restrictions at the weekend in SotG and the surrounding roads. As of last weekend there is, in effect, no weekend parking for non-residents between Kew Bridge and the Railway Bridge, and the adjoining roads.
But you can still park on SotG and adjoining roads south-east of the Railway Bridge (Magnoliia, Loraine, Riverview etc.) at the weekend.