It looks like you misread slightly: partial orders and equivalence relations are both reflexive and transitive, but only equivalence relations are symmetric, while partial orders are antisymmetric. A relation $R$ is symmetric if $aRb$ implies $bRa$, while a relation $R$ is antisymmetric if $aRb$ and $bRA$ implies $a=b$.
For example, the relation "has the same parity as" is symmetric (and in fact an equivalence relation) on the integers, since if $a$ has the same parity as $b$ then $b$ has the same parity as $a$. On the other hand, it is not antisymmetric, because $2$ has the same parity as $4$ and $4$ has the same parity as $2$, even though $2\neq 4$.
The relation $\leq$ on integers, in contrast, is antisymmetric (and in fact a partial order), since if $a\leq b$ and $b\leq a$ then $a=b$. But it is not symmetric, since (for instance) $2\leq 4$ but $4\not\leq 2$.