We may assume the sides are either parallel to the $x$- or $y$-axes and that the lowest horizontal sides are at the $x$-axis. Start with a horizontal side that is on the $x$-axis, go around the polygon in clockwise direction, label the horizontal sides $1, 2, \ldots, 1008$ and put an arrow on the side along the direction that you proceed. Each horizontal side $s_i$ forms a rectangle with the $x$-axis whose height is the height of $s_i$ above the $x$-axis and whose width is $s_i$. If the arrow on $s_i$ points in the direction positive $y$-axis, its area $A_i$ is positive. Otherwise it is negative. The area of $P$ is the absolute value of
$$A_1 + A_2 + \dots + A_{1008}.$$
Since the side lengths are all odd, $A_i$ and $A_{i+1}$ for $i = 1, \dots, 1007$, have opposite parities. Hence there are $504$ terms which are odd. Therefore the area is even.