There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one-and two-component FQH systems at total filling fraction nu = n + 2/3, for integer n. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction nu = n + 2/3, including in particular the possibility of the non-Abelian Z(4) parafermion state. In nu = 2/3 bilayers we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the Z(4) state. On the other hand, in single-component systems at nu = 8/3, we find that the Z(4) parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed nu = 8/3 state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.