Let $S$ and $T$ be elliptic differential operators on vector bundles $E\rightarrow M$ and $F\rightarrow N$ respectively, where $M$ and $N$ are manifolds. Suppose $Q_S$ and $Q_T$ are parametrices for $S$ and $T$. Form the operator $S\#T$ on the external product bundle $E\boxtimes F\rightarrow M\times N$, defined as $$S\#T=S\otimes 1+1\otimes T.$$

Then $S\#T$ is an elliptic operator.

Question: Is there an easy way to write down a parametrix $Q_{S\#T}$ for $S\#T$ in terms of $Q_S$ and $Q_T$? If so, what would the Schwartz kernel of $Q_{S\#T}$ look like?

Thanks!