That paracompact Hausdorff implies normal is standard and there are examples on StackExchange of perfectly normal Hausdorff spaces that are not paracompact, but I'm not sure of the answer, especially since paracompact spaces are collection-wise normal and the latter is not related to perfectly normal. (The standard example of a collection-wise normal space that is not perfectly normal is $\omega_1$, which is not paracompact.)
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No. Consider the closed ordinal space $\omega_1+1 = [0,\omega_1]$. Since it is compact, it is clearly paracompact, and is also Hausdorff. But it is not perfectly normal: the (closed) singleton $\{ \omega_1 \}$ is not a Gδ subset.
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1Thanks! So even a compact hausdorff space need not be perfectly normal. – chrystomath Jun 24 '14 at 10:51
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1@chrystomath: You're welcome! Yes, compact Hausdorff spaces need not be perfectly normal. In fact, they may fail to be even hereditarily normal: for example, the Tychonoff plank. – user642796 Jun 24 '14 at 11:31