There are several theorems of Ernest Michael from 1950s about paracompactness. There is also Michael’s 1972 characterization of paracompact locally compact spaces under certain class of quotient maps.
Theorem. (Michael theorem on the closed image of a paracompact) An image of a paracompact Hausdorff space under a closed continuous map is also paracompact Hausdorff.
Michael selection theorem: a lower semicontinuous map from a paracompact topological space $X$ to a Banach space $E$ with convex closed values has a continuous subrelation which is a function. If this is true for a given topological space $Y$ instead of $E$ and all such functions and codomains $E$, then $Y$ is paracompact.
Ernest Michael, Proc. Amer. Math. Soc. 1953
E. Michael, Continuous selections. I, Annals of Math. 63 (2): 361–382, doi, MR0077107
E, Michael, Another note on paracompactness, 1957, pdf
E. A. Michael, A quintuple quotient test, Gen. Topol. Appl. 2 (1972) 91–138, link
discussion at A. Methew’s blog and also on an application to metric spaces here
R. Engelking, General topology
Kenneth Kunen, Paracompactness of box products of compact spaces, Trans. Amer. Math. Soc. 240, (1978) 307-316, jstor
wikipedia Michael selection theorem