The Weierstrass $\wp$ function with an associated lattice $L$ is given by the following equation for $z \notin L:$ $$ \wp(z)=\frac{1}{z^{2}}+\sum_{\omega \in L \backslash\{0\}}\left[\frac{1}{(z-\omega)^{2}}+\frac{1}{\omega^{2}}\right]. $$ What we want to show that $\wp(u) = \wp(v)$ if and only if $u-v$ or $u+v$ is a period of $\wp$.
My attempt:
($\Leftarrow$): If $u-v$ is a period of $\wp$, then $\wp((u-v)+z)=\wp(z)$. In particular, we have that $$\wp(u)=\wp((u-v)+v)=\wp(v).$$ Similarly, If $u+v$ is a period of $\wp$, then $\wp((u+v)+z)=\wp(z)$. In particular, we have that $$\wp(u)=\wp((u+v)-v)=\wp(-v)=\wp(v),$$ since $\wp$ is an even funciton.
($\Rightarrow$): Is there any idea to show this direction?
COMMENT.- The attached figure shows a fundamental parallelogram and the values that the Wierstrass function takes in it. Can you deduce from this the properties that you enunciate?