The unexpected appearance of a fractional quantum Hall effect (FQHE) plateau at nu = 2 + 6/13 [A. Kumar et al., Phys. Rev. Lett. 105, 246808 (2010)] offers a clue into the physical mechanism of the FQHE in the second Landau level (SLL). Here we propose a "(3) over bar(2) over bar 111" parton wave function, which is topologically distinct from the 6/13 state in the lowest Landau level. We demonstrate the (3) over bar(2) over bar 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 "(n) over bar(2) over bar 111" family of parton states naturally describes many observed SLL FQHE plateaus.