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Denniston constructed partial difference sets (PDSs) with the parameters$(2^{3m}, (2^{m+r} - 2^m + 2^r)(2^m-1), 2^m-2^r+(2^{m+r}-2^m+2^r)(2^r-2),(2^{m+r}-2^m+2^r)(2^r-1))$ in elementary abelian groups of order $2^{3m}$ forall $m \geq 2, 1 \leq r < m$. In this paper, we show that PDSs with theDenniston parameters $(p^{3m}, (p^{m+r} - p^m + p^r)(p^m-1),p^m-p^r+(p^{m+r}-p^m+p^r)(p^r-2), (p^{m+r}-p^m+p^r)(p^r-1))$ exist in allelementary abelian groups of order $p^{3m}$ for all $m \geq 2, r \in \{1,m-1\}$ where $p$ is an odd prime, and present a construction. Our approach usesPDSs formed as unions of cyclotomic classes.

Partial difference sets with Denniston parameters and cyclotomy