The distance between two skew lines is given by the equation
$$d = \frac{| (\vec{a}_2 – \vec{a}_1) . (\vec{b}_1 \times \vec{b}_2) |} {| \vec{b}_1 \times \vec{b}_2|}$$ Basically, the equation finds the projection of $PQ$ on $AB$. But why can't we apply the same formula if the two lines are parallel? If we apply the same logic of finding the projection of $PQ$ on $AB$ we should get the formula for distance between parallel lines to be $$d = \frac{| (\vec{a}_2 – \vec{a}_1).\vec{b} |} {| \vec{b}|}$$ where $\vec{b}$ is the vector perpendicular to the parallel lines.
However, the distance between parallel lines is given by the equation $$ d = \frac{|(\vec a_{2}-\vec a_{1})×\vec b|}{|\vec b|}$$ where $\vec{b}$ is the vector parallel to the parallel lines.
Edit
I think both formulas are correct, except that it is not straightforward to find a vector perpendicular to the parallel lines and passing through both of them. Am I correct?
If $b_2=\lambda b_1$ (they are parallel) then the numerator of the formula is: $$\|(a_2-a_1)\cdot(b\times\lambda b)\|=\det((a_2-a_1)\,\,b\,\,\lambda b)=0$$And the denominator is also zero. The formula is useless for parallel vectors as the magnitude of the cross prodduct is nil.
The quantity: $$\frac{\|(a_2-a_1)\cdot b\|}{\|b\|}$$Does not give the distance between the lines (unless by accident). It rather gives you the length of the projection of the vector $a_1\to a_2$ onto $b$, as it is $\|a_2-a_1\|\cdot|\cos\theta|$ for the angle $\theta$ between them. You want the sine for the distance instead, which is what you get with the cross product.