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.