The distance between two parallel lines in 3D space can be calculated by finding the perpendicular distance between a point on one line and the other line. If we have two parallel lines with direction vector d, and the lines pass through points P₁ and P₂, then the formula for the distance D is:

D = |(P2 - P1) × d| / |d|
Here, P₂-P₁ is the vector between any two points on the lines, and the cross product gives the area of the parallelogram formed by the two vectors. Dividing by the magnitude of the direction vector d yields the shortest distance between the lines.