Let $X$ be a set and let ${\frak T}$ be a collection of paracompact topologies on $X$ such that for any $\tau, \tau'\in {\frak T}$ we have $\tau\subseteq \tau'$ or $\tau'\subseteq \tau$. It is easy to see that $\bigcup {\frak T}$ is a topology.

Is $(X,\bigcup {\frak T})$ necessarily paracompact?

Let $X$ be a set and let ${\frak T}$ be a collection of paracompact topologies on $X$ such that for any $\tau, \tau'\in {\frak T}$ we have $\tau\subseteq \tau'$ or $\tau'\subseteq \tau$. Let $\sigma$ be the topology having $\bigcup {\frak T}$ as a base.

Is $(X,\sigma)$ necessarily paracompact?

EDIT. Thanks to Tomek Kania for observing that $\bigcup {\frak T}$ need not be a topology.