This one is probably simple, but I don't see it yet.
Is a locally compact, pseudocompact Hausdorff space necessarily paracompact?
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Any pseudocompact paracompact Hausdorff space is compact. So $\omega_1$ with the order topology is a counterexample to your question since it is locally compact, pseudocompact (any real valued continuous function is eventually constant) and not compact.
The long line is one of the standard examples of a space that is not paracompact. It is however Hausdorff, locally compact, and pseudocompact because maps from it to $\mathbb R$ are eventually constant.