The two sets of $m$ parallel lines along with two sets of two parallel lines of the given parallelogram will form two sets of $\left(m+2\right)$ parallel lines. Each parallelogram is formed by choosing two parallel lines from each of the above

$\therefore$ Total number of parallelograms

$={}^{m+2}C_2\times {}^{m+2}C_2={\left({}^{m+2}C_2\right)}^2$

Hence, option (c) is correct.