A parallelogram is cut by two sets of m lines parallel to its sides.

Each set of parallel lines consists of $(m+2)$ lines.

Each parallelogram is formed by choosing two lines from the first set and two straight lines from the second set.

$\therefore$ Total number of parallelograms $={ }^{m+2} C_{2} \times{ }^{m+2} C_{2}=\left({ }^{m+2} C_{2}\right)^{2}$