Can the perimeter of a convex polygon X be an odd number?

I suspect you can drop "convex" and the first two constraints and it is still true that the parity of the sum is even

Consider your polygon has sides length $z_1,z_2,\ldots,z_n$ where $z_i^2=x_i^2+y_i^2$ and $\sum x_i = \sum y_i = 0$. The $x_i$ and $y_i$ are the signed differences in co-ordinates between successive vertices, corresponding to the horizontal and vertical components of each side. Everything (the lengths, the co-ordinates and the difference in co-ordinates) is an integer.

Then the parity of $z_i$ is

• equal to the parity of $z_i^2$
• equal to the parity of $x_i^2+y_i^2$
• equal to the parity of $x_i+y_i$

so the parity of the circumference $\sum z_i$ is equal to the parity of $\sum x_i + \sum y_i$, which being $0$ is even.