Poly-time reduction is not antisymmetric
By definition, a relation $R$ is antisymmetric if $aRb$ and $bRa$ implies $a=b$. Therefore, to show that $R$ is not antisymmetric, we have to find $a \neq b$ such that $aRb$ and $bRa$.

Specializing this to your case, you have to find two different languages $L_1 \neq L_2$ such that $L_1 \leq_p L_2$ And $L_2 \leq_p L_1$. Finding a pair of such languages isn't too challenging, so I leave it to you.