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.