What is the ATC phraseology to request a parallel track?
It is rather simple:


  OE-DLM, request 2NM right of track. 


However, weather circumnavigation is mostly done with headings. This way it is easier for the controller to predict your movement and thus preferred by many:


  OE-DLM, request heading 120 for the next 20NM to avoid (weather).


This way it is easy for the controller to predict your movement and he has an idea where your avoidance will be completed. Additionally you are not forced to turn behind a cell you weren’t able to look behind before. 

Mainly done in oceanic airspace where you try to make your way through the ITCZ (InnerTropical Convergence Zone) is this variant:


  OE-DLM, request up to 20NM right of track to avoid (weather). 


If approved you got a 20NM wide corridor to play in. 

Flying offset is rather uncommon in the crowded European airspace (or generally every crowded airspace) but a normal thing everywhere else. China even uses it as a tactical separation method employed by the controller.