Exercise
A space $X$ is said to be $\sigma$-compact if it can be written as a countable union of compact subspaces. Is the product of a paracompact space $X$ with a $\sigma$-compact space $Y$ always paracompact? (In this case, compact and paracompact need not be Hausdorff as part of definition)
Partial Answer
If $X\times Y$ is regular, then it is paracompact.
Indeed, if a space is regular and every open cover has a $\sigma$-locally finite (i.e. a countable union of locally finite families) open refinement, then it is paracompact.
Let $Y$ be the union of compact subspaces $Y_1,Y_2,Y_3,\dots$. Let $\mathcal{C}$ be a family of basic open sets such that $X\times Y=\bigcup\mathcal{C}$. For $n$, for $x\in X$, there are $W_{nx1},\dots,W_{nxm_{nx}}\in\mathcal{C}$ such that $\{x\}\times Y_n\subseteq W_{nx1}\cup\cdots\cup W_{nxm_{nx}}$ and, if $W_{nxi}=U_{nxi}\times V_{nxi}$, then $x\in U_{nxi}$. Therefore $Y_n\subseteq V_{nx1}\cup\cdots\cup V_{nxm_{nx}}$. If $U_{nx}=U_{nx1}\cap\cdots\cap U_{nxm_{nx}}$, then $U_{nx}$ is open and $x\in U_{nxi}$. Then there is a family $(A_{ns})_{s\in S_n}$ of open sets such that $X=\bigcup_{s\in S_n}A_{ns}$, for $s\in S_n$ there is $x_s\in X$ such that $A_{ns}\subseteq U_{nx_s}$ and for every $x\in X$ there is an open set $T$ such that $x\in T$ and $\{s\in S_n\mid T\cap A_{ns}\neq\emptyset\}$ is finite. Then $(A_{ns}\times V_{nx_si})_{n\geq1, s\in S_n, 1\leq i\leq m_{nx_s}}$ is a family of open sets. We have $X\times Y=\bigcup_{n=1}^\infty(X\times Y_n)=\bigcup_{n=1}^\infty\bigcup_{s\in S_n}\bigcup_{i=1}^{m_{nx_s}}(A_{ns}\times V_{nx_si})$. For every $n\geq1$, $s\in S_n$ and $1\leq i\leq m_{nx_s}$ then $A_{ns}\times V_{nx_si}\subseteq W_{nx_si}$. For $n\geq1$, then for every $(x,y)\in X\times Y$ there is an open set $T$ such that $x\in T$ and $S'=\{s\in S_n\mid T\cap A_{ns}\neq\emptyset\}$ is finite, so $T\times Y$ is open, $(x,y)\in T\times Y$ and $\{(s,i)\mid s\in S_n,1\leq i\leq m_{nx_s}, (T\times Y)\cap(A_{ns}\times V_{nx_si})\neq\emptyset\}\subseteq\bigcup_{s\in S'}(\{s\}\times\{1,\dots,m_{nx_s}\})$ is finite. Therefore the family $(A_{ns}\times V_{nx_si})_{n\geq1, s\in S_n, 1\leq i\leq m_{nx_s}}$ is a family of open sets which covers $X\times Y$, is a refinement of $\mathcal{C}$ and is $\sigma$-locally finite. Therefore $X\times Y$ is paracompact.
Question
What about if $X\times Y$ is not regular?