Parallelogram $ABCD$ has area $1296$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y=x$ and $y=kx$ for some integer $k>1$. How many such parallelograms are there?

1. With $B=(b,b)$ and $D=(d,kd)$, the area is $bd(k-1)=N$, so we must count ordered triples of positive integers $(b,d,k-1)$ with product $N$.
2. $N=1296=2^4\cdot3^4$. For a single prime with exponent $e$, the number of ways to write it as an ordered product of $3$ positive integer factors is $\binom{e+2}2$; here $\binom62=15$ for each prime.
3. Total: $15\times15=\boxed{225}$.