smc

The area of any parallelogram $ABCD$ can be computed as the size of the vector product of $\overrightarrow{AB}$ and $\overrightarrow{AD}$. In our setting where $A=(0,0)$, $B=(s,s)$, and $D=(t,kt)$ this is simply $s\cdot kt-s\cdot t=(k-1)st$. In other words, we need to count the triples of integers $(k,s,t)$ where $k>1$, $s,t>0$ and $(k-1)st=1,000,000=2^6{5}^6$. These can be counted as follows: We have $6$ identical red balls (representing powers of $2$), $6$ blue balls (representing powers of $5$), and three labeled urns (representing the factors $k-1$, $s$, and $t$). The red balls can be distributed in $\left(\genfrac{}{}{0ex}{}{8}{2}\right)=28$ ways, and for each of these ways, the blue balls can then also be distributed in $28$ ways. (See Distinguishability for a more detailed explanation.) Thus there are exactly $28^2=784$ ways how to break $1,000,000$ into three positive integer factors, and for each of them we get a single parallelogram. Hence the number of valid parallelograms is $784⟶\boxed{784}$.