Answer
$r = u\sec \left( {\theta - \frac{\pi }{3}} \right)$, where $u$ is any real number.
Work Step by Step
Since $r\cos \left( {\theta - \frac{\pi }{3}} \right) = 1$ is a line, the polar equation can be written as
$r = \frac{1}{{\cos \left( {\theta - \frac{\pi }{3}} \right)}} = \sec \left( {\theta - \frac{\pi }{3}} \right)$.
The slope of this line is given by $\tan \theta = \frac{y}{x} = \frac{{r\sin \theta }}{{r\cos \theta }}$.
Notice that we can scale $r$ by multiplying it with any number such that the slope does not change. Thus, we obtain a family of lines parallel to the original line by multiplying $r$ with a constant. Hence, the polar equations of the lines parallel to the line $r = \sec \left( {\theta - \frac{\pi }{3}} \right)$ is just
$r = u\sec \left( {\theta - \frac{\pi }{3}} \right)$, where $u$ is any real number.
