In this question I will understand the definition of paracompactness to include the Hausdorff separation axiom.
Let $X$ be a paracompact space and $A\subseteq X$ a closed subspace. Let $f:A\rightarrow B$ be a continuous map to a paracompact space $B$ and form the adjunction space $B\cup_fX$ as the pushout in the next square $\require{AMScd}$ \begin{CD} A@> \subseteq >> X\\ @VfV V @VV V\\ B @>>> B\cup_fX. \end{CD} I believe it is true that $B\cup_fX$ is paracompact under the assumptions above. The proof of this that I have seen used Michael's Selection Theorem.
Is there a more direct way to prove that $B\cup_fX$ is paracompact? Say using only the machinery developed in Munkres's text book?
In this classic paper E.Michael showed that the closed and continuous image of a paracompact space is again paracompact. He does this via a characterisation of paracompactness that stays pretty close to how Munkres treats paracompactness, so it should be pretty accessible to you.
And in your set-up, the (quotient) map from the sum of $B$ and $X$ (which is paracompact) to $B\cup_f X$ is closed (as $A$ is closed in $X$).