  We investigate the nature of the fractional quantum Hall (FQH) state at
filling factor $\nu=13/5$, and its particle-hole conjugate state at $12/5$,
with the Coulomb interaction, and address the issue of possible competing
states. Based on a large-scale density-matrix renormalization group (DMRG)
calculation in spherical geometry, we present evidence that the physics of the
Coulomb ground state (GS) at $\nu=13/5$ and $12/5$ is captured by the $k=3$
parafermion Read-Rezayi RR state, $\text{RR}_3$. We first establish that the
state at $\nu=13/5$ is an incompressible FQH state, with a GS protected by a
finite excitation gap, with the shift in accordance with the RR state. Then, by
performing a finite-size scaling analysis of the GS energies for $\nu=12/5$
with different shifts, we find that the $\text{RR}_3$ state has the lowest
energy among different competing states in the thermodynamic limit. We find the
fingerprint of $\text{RR}_3$ topological order in the FQH $13/5$ and $12/5$
states, based on their entanglement spectrum and topological entanglement
entropy, both of which strongly support their identification with the
$\text{RR}_3$ state. Furthermore, by considering the shift-free
infinite-cylinder geometry, we expose two topologically-distinct GS sectors,
one identity sector and a second one matching the non-Abelian sector of the
Fibonacci anyonic quasiparticle, which serves as additional evidence for the
$\text{RR}_3$ state at $13/5$ and $12/5$.
