Henno Bransma provided this answer in a comment to anon's answer, which I think is the easiest and best, hence I'm adding it as a community wiki answer.
Consider an uncountable set $\mathfrak{n}$ and the space $\mathbb{R}^\mathfrak{n}$ in the product topology. This is not normal, but still Hausdorff, hence not paracompact.
