I've lately came to know that the axiom $\sf V=HOD$ can be stated as: $$\forall X \, \exists \theta \, \exists \alpha < \theta \, \exists \varphi : X=\{y \in V_\theta\mid V_\theta\models\varphi(y,\alpha)\}$$ Now my question: is it possible along the same general lines to state the following axiom: $$\forall X \, \exists \theta \, \exists p_1,..,p_n \in V_\theta \, \exists \varphi : \\ X=\{y \in V_\theta\mid V_\theta\models\varphi(y,p_1,..,p_n)\}$$ Call that as $\sf V= HPD$, for Hereditarily Parameteric Definable.
Notice that here the parameters $p_1,..,p_n$ in the above axiom are not restricted to being ordinals.
If that is possible, then is it equivalent to $\sf V=HOD$ over the rest of axioms of $\sf ZF$?
I claim that $\sf ZF$ already proves $\sf V=HPD$. Moreover, it is enough to restrict ourselves to a single atomic formula: $\varphi(y,p):=y\in p$.
For every $X$, there exists a $\theta$ such that $X\in V_\theta$, so we may take $p=X$, and then we have, with our chosen $\varphi$ $$X=\{y\in V_\theta\mid V_\theta\models y\in X\}.$$
So this axiom is most certainly not equivalent to $\sf V=HOD$.