Yes, because there is a definable ordinal pairing function.
Specifically, if you want to get the set $\{y\in V_\theta\mid V_\theta\models\varphi(y,\alpha)\}$, then let $\beta=\langle\theta,\alpha\rangle$ be the ordinal coding the pair, and then look at $V_{\beta+1}$. Inside this structure, we have $\beta$ as the largest ordinal, and the model can decode $\beta$ as the pair $\langle\theta,\alpha\rangle$, and then get the set $\{y\in V_\theta\mid V_\theta\models\varphi(y,\alpha)\}$. So this set is definable inside $V_{\beta+1}$ without any extra parameter.
Ultimately one should view $\varphi$ also as a parameter (and formulas can be coded by finite numbers, hence by ordinals, a fact that is important when proving that HOD as defined internally is the same as the external concept of definable-from-ordinal parameters, since the internal definition might use nonstandard $\varphi$, but this is still OK since the code of a nonstandard formula is still a number, hence an ordinal). But this is fine, since we can consider $\beta=\langle\theta,\alpha,\ulcorner\varphi\urcorner\rangle$ and then proceed as I explained.