"It looks like one cuts across a large park to walk between Old Town and New Town."
No, this is the "everything is uphill" that was referred to above. That park is lovely, but sunken, as well as filled with winding paths. It's a great diversion, but it would take you an unnecessarily long time and involve extra effort if you were using it merely to connect the Old and New Towns.
"I assume people make the 20-minute walk from Old Town back to a hotel or apartment in New Town?"
You can, but I mostly took the bus (again, hills and distance and time). A day pass on the buses, at least in 2018, was only £4, and the buses are frequent between these two points, as they are served by multiple routes (so, I just took the first one that arrived). Note that the buses only take coins and do not give change; I believe you can also get tickets on the phone app, but don't have details.