  The Read-Rezayi (RR) parafermion states form a series of exotic non-Abelian
fractional quantum Hall (FQH) states at filling $\nu = k/(k+2)$.
Computationally, the wave functions of these states are prohibitively expensive
to generate for large systems. We introduce a series of parton states, denoted
"$\bar{2}^{k}1^{k+1}$," 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 $(U(1)_{k+1} \times U(2k)_{-1})/(SU(k)_{-2} \times U(1)_{-1})$
coset conformal field theory describes the edge excitations of the
$\bar{2}^{k}1^{k+1}$ state, suggesting non-trivial 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 "$\bar{n}\bar{2}^{2}1^{4}$," with $n=1,2,3$, to describe
the FQH states observed at $\nu=2+1/2,~2+2/5$ and $2+3/8$.
