The beautiful answer given by Alexander can be consider as a particular case of the general framework developed in AM16, specifically in its appendix A.
The objects of the parity complex of cubes in $\infty$-$\mathcal{C}at$ can be defined inductively as the tensor products $\Delta_1 \otimes \dots \otimes \Delta_1$ and they are free objects (in the sense of polygraph/computads), see remark A.16. So it is more natural to define at first the general Gray tensor product for $\infty$-$\mathcal{C}at$ and then truncate to get a Gray tensor product in any degree $n$.
Let $\mathcal{C}_{\text{da}}$ be the category of directed augmented complex and $\mathcal{S}t_{\text f}$ its full subcategory of Steiner complexes, see ch. 2 of AM16. There exists a functor $\nu \colon \mathcal{C}_{\text{da}} \to \infty$-$\mathcal{C}at$ which by a theorem of Steiner is fully faithful when restricted to $\mathcal{S}t_{\text f}$, so that we call the objects of the essential (restricted) image Steiner $\infty$-categories. Another result by Steiner ensures that Steiner complexes are closed under tensor product of directed augmented complexes, see proposition A.4. Moreover, the orientals, the cubes and $\Theta$ are Steiner complexes/categories. Theorem A.14 tells us that the Gray tensor product on $\infty$-$\mathcal{C}al$ is precisely the left Kan extension
$$ \require{AMScd} \begin{CD} \Theta \times \Theta @>\nu\times \nu>> \infty\text{-}\mathcal {C}at\times\infty\text{-}\mathcal{C}at\\ @V\otimes VV @VV\otimes_{\text{Gray}} V\\ \mathcal{S}t_f @>>\nu> \infty\text{-}\mathcal{C}at \end{CD}$$
Finally, by lemma A.25 and proposition A.26, we get the 2-categorical Gray tensor product as the following left Kan extension:
$$ \require{AMScd} \begin{CD} \Theta_2 \times \Theta_2 @>\nu\times \nu>> 2\text{-}\mathcal {C}at\times 2\text{-}\mathcal{C}at\\ @V\otimes VV @VV\otimes_{\text{Gray}} V\\ \mathcal{S}t_f @>>\tau^1_{\leqslant 2}\circ \nu> 2\text{-}\mathcal{C}at \end{CD}$$
The role of $\Theta$ and $\Theta_2$ is that they are small dense sub-categories satisfying a bunch of nice properties, see theorem 6.3. Role which can be played by the 2-trucation of the cubes $\mathbf{Cu}$ for the second case of $2\text{-}\mathcal{C}at$. I don't know whether the full subcategory of cubes is dense in $\infty$-$\mathcal{C}at$, though.