A taxiway does not have to be parallel with the runway - it is most certainly NOT parallel at BOH for instance.
There is room to extend the taxiway to the 20 end without having to realign roads, etc. The "queuing" area at the 27 end at London City has a centreline only 85m off the runway centreline. There is easily more space than this at SOU and the vast majority of aircraft using SOU are of a similar size to those at LCY.
Even if they did not extend the taxiway surely they could build a connection to the runway at the north end of the GA apron to halve the backtracking distance?