Are these 2 equivalent?
It does hold that
$$
\forall x \forall y \forall z \, [p(y,z) \to p(x,z)] \quad \longleftrightarrow \quad
\forall x \forall y \forall z \, [p(x,z) \to p(y,z)],
$$
since $\forall x \forall y$ is the same thing as $\forall y \forall x$. So your logic is incorrect.

Similarly, if $\varphi$ is promised to be symmetric then the equivalence holds, for similar reasons.

A simple example where the equivalence doesn't hold is $\varphi(x,y) = x$, $p(x,y) = x$.

In this case the first statement is
$$
\forall x \forall y \forall z \, [x \land y \to x],
$$
whereas the second statement is
$$
\forall x \forall y \forall z \, [x \land x \to y].
$$
(This assumes that the statements are interpreted as $(\varphi \land p) \to p$ rather than $\varphi \land (p \to p)$.)