  The unexpected appearance of a fractional quantum Hall effect (FQHE) plateau
at $\nu=2+6/13$~ [Kumar \emph{et al.}, Phys. Rev. Lett. {\bf 105}, 246808
(2010)] offers a clue into the physical mechanism of the FQHE in the second
Landau level (SLL). Here we propose a "$\bar{3}\bar{2}111$" parton wave
function, which is topologically distinct from the 6/13 state in the lowest
Landau level. We demonstrate the $\bar{3}\bar{2}111$ state to be a good
candidate for the $\nu=2+6/13$ FQHE, and make predictions for experimentally
measurable properties that can reveal the nature of this state. Furthermore, we
propose that the "$\bar{n}\bar{2}111$" family of parton states naturally
describes many observed SLL FQHE plateaus.
