There is exist tradition to formulate axiom schemes such as induction, comprehension or replacment in a way like "for each formula $\phi(y,\overline{x})$ with free variables $y$ and $\overline{x}=x_1,...,x_n$ we have $\forall \overline{x} ( \text{text of axiom scheme which include } \phi(y,\overline x))$". To not be vague, let take as archetypical example comprehension scheme for second order arithmetics from ncatlab article.
$ \forall \overline m \forall \overline X \exists Z \forall n (n \in Z \leftrightarrow \phi(n,\overline m, \overline X))$
I choose that example because I don't know it is equivalent to axiom without parameters or not, whenever equivalence to $ZFC$ or first order $PA$ to versions without parameters is classical result. I understand that apriori allowing parameters could increase expressible power of our system, but I don't understand why we not allowing not only $\Pi_1$ parameters but parameters of all type like, for example, next $\Sigma_3$ axiom scheme
$\exists \overline m_1 \exists \overline X_1 \forall \overline m_2 \forall \overline X_2 \exists \overline m_3 \exists \overline X_3 \exists Z \forall n (n \in Z \leftrightarrow \phi(n,\overline m_1, \overline X_1, \overline m_2, \overline X_2, \overline m_3, \overline X_3))$
it seems for me like it could increase our expressible power a lot, so, why it is not traditional?