the shortest distance between the two parallel lines will be on a line that is perpendicular to both (and that passes through both).
You can use this to get the coordinates of 2 points that are at the minimum distance from each other, find the distance between those two points to get answer you want.
Possibly Shorter way (requires calculus):
Pick a point on one of your lines
Note that every point on your second line satisfies where a is a constant, the m's are the slope paramters of your second line, and is a known point on your second line
Setup a lagrangian to minimise the distance between and by choosing a.
This minimum distance is the distance you want.