The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian fractional quantum Hall (FQH) states at filling. = k/(k + 2). Computationally, the wave functions of these states are prohibitively expensive to generate for large systems. We introduce a series of parton states, denoted ($2) over bar (k)1(k+1), and show that they lie in the same universality classes as the particle-hole-conjugate RR ("anti-RR") states. Our analytical results imply that a [U(1)(k+1) xU(2k)(-1)]/[SU(k)(-2) xU(1)(-1)] coset conformal field theory describes the edge excitations of the (2) over bar (k)1(k+1) state, suggesting nontrivial dualities with respect to previously known descriptions. The parton construction allows wave functions in anti-RR phases to be generated for hundreds of particles. We further propose the parton sequence (n) over bar(2) over bar (4), with n = 1, 2, 3, to describe the FQH states observed at nu= 2 + 1/2, 2 + 2/5, and 2 + 3/8.