In order to apply the metric to lines, you likely consider these lines as sets of points. Two lines intersect if they have a point in common, ore phrased in terms of the metric, if there exists a point on the first line and a point on the second line which have distance zero to one another.
If you have a pair of parallel lines, they don't intersect, so for any point $A$ on the first line and $B$ on the second line, the distance will be non-zero. If you apply a rigid motion to them, the corresponding image points will still have that property, so you don't get any at distance zero. Therefore the resulting lines must still be parallel.
You can generalize this to say that rigid motions preserve incidence, since two sets of points are incident if they have a point in common.
If you care about the distinction between asymptotically parallel lines ans ultraparallel lines, as your comment indicates, then you'd observe that the asymptotically parallel lines are those of the set of all parallel lines which are closest to a given line. If distances are preserved, then so is the property of being closest. Together with the above this guarantees the preservation of asymptotically parallel lines.