I have given both lines an offset to make them appear parallel to each other, but I cannot get them to connect to the main line properly. Unfortunately, I cannot currently post more images to describe my problem as I lack the reputation to do so.
I guess you want something similar to the network map and Dr.Lluch Area map from http://www.metrovalencia.es/page.php?page=145.
I'm afraid there is no proper solution for that with QGIS. You can split multipart lines to single part and change line directions to exchange the placement of the lines, but you will always get some nasty crossings (as the Dr.Lluch map has in the south west edge).
This is the best I got from OpenStreetMap data, with a line width of 2 mm and offset +/- 1mm:
At that point, a CAD solution will always look better than real GIS drawings.