Let $m = 504$ and $p_0, p_1, \ldots, p_{4m-1}$ be the $2016$ vertices of the polygon $\Delta$ ordered in counterclockwise manner. Let $p_{4m} = p_0$ and $(x_k,y_k)$ be the coordinates of $p_k$. WOLOG, we will assume all $x_k, y_k$ are integers and the edges are either horizontal or vertical.
Cyclic reordering if necessary, we will assume $y_0 = \min \{ y_k \}$
and $x_0 = \min \{ x_k : y_k = y_0 \}$.
Shift the origin so that $p_0 = (0,0)$. It is easy to see for all $k$, the parity of coordinates satisfy:
$$p_k = (x_k,y_k) \equiv
\begin{cases}
(0,0), & k \equiv 0 \pmod 4\\
(1,0), & k \equiv 1 \pmod 4\\
(1,1), & k \equiv 2 \pmod 4\\
(0,1), & k \equiv 3 \pmod 4
\end{cases}$$
Using Green's theorem, we can evaluate the area of $\Delta$ as a line integral over its boundary $\partial \Delta$:
$$\verb/Area/(\Delta) = \int_\Delta dx dy = \int_{\partial \Delta} x dy$$
Since $\partial\Delta$ are a bunch of horizontal or vertical segments, we can convert the last integral to a sum over corresponding edges${}^{\color{blue}{[1]}}$.
We can split the 2016 edges of the polygon into $4$ groups.
- $p_{4\ell} \to p_{4\ell+1}$ : doesn't contribute because this is a horizontal edge.
- $p_{4\ell+1} \to p_{4\ell+2}$ : $x$ is constant and an odd number. Since $y_{4\ell+2} - y_{4\ell+1}$ is also odd, this contributes an odd number to area.
- $p_{4\ell+2} \to p_{4\ell+3}$ : doesn't contribute because this is a horizontal edge.
- $p_{4\ell+3} \to p_{4\ell+4}$ : $x$ is constant and an even number.
Since $y_{4\ell+4} - y_{4\ell+3}$ is an integer, this contributes an even number to the area.
This means in general, we have
$$\int_{p_{k} \to p_{k+1}} xdy \equiv \begin{cases}
1 \pmod 2, & k \equiv 1 \pmod 4\\
0 \pmod 2, & k \not\equiv 1 \pmod 4
\end{cases}$$
Summing $k$ from $0$ to $4m-1$, we obtain
$$\verb/Area/(\Delta) = \sum_{k=0}^{4m-1}
\int_{p_{k} \to p_{k+1}} xdy \equiv m \pmod 2$$
Since $m$ is an even integer, so does the area of original polygon $\Delta$.
Notes
- $\color{blue}{[1]}$ - For those who don't want to use calculus, one can replace the line integral by following specific form of shoelace formula:
$$\verb/Area/(\Delta) = \left|\sum_{k=0}^{4m-1} \frac{x_k + x_{k+1}}{2}(y_{k+1} - y_k)\right|$$
The rest of arguments will be exactly the same.