Hello loggerhead
Our airline shuts down the right engine on the way IN, -- never out.
As JBravo states, it has a lot to do with brake wear - and fuel savings. The 737 (300-400-500) will accelerate in idle when light after flight.(not the 600-700) Our ground staff always stands ready at the gate with the ground power plug, and I can then shut down after setting the park brake. We do not (normaly)use the APU either. It works just fine, and keep the engineers happy. BUT it's not mandatory. A winter day demands different thinking.