Q: What does mean "run parallel to"? In the following sentence, what is the meaning of "runs parallel to"?

The street where I live runs parallel to the main road through town.

Is it a phrasal verb? (I could not find it in any dictionary or internet resources!)

A: 
run (v): 4. Extend or cause to extend in a particular direction.  cobbled streets run down to a tiny harbour
parallel (adj): 1. (of lines, planes, or surfaces) side by side and having the same distance continuously between them. parallel lines never meet

The combined meaning is that the roads "extend in a particular direction, with a parallel relationship to each other".

A: streets are said to "run" east-west or north-south, for example. Run here just means "is".
Many things can be said to run in reference to the real world: the property line runs close to the street. [is close to]. 
The street where I live is parallel to x. Same thing as "runs parallel to".
Lines, streets, etc. run in this or that direction or are parallel or perpendicular to something.
