Understanding the links between different phases of outflows from active galactic nuclei is a key goal in extragalactic astrophysics. Here, we compare [OIII] {lambda}{lambda} 4960, 5008 outflow signatures in quasars with and without broad absorption lines (BALs), aiming to test how the broad absorption troughs seen in the rest-frame ultraviolet are linked to the narrow line region outflows seen in the rest-frame optical. We present new near-infrared spectra from Magellan/FIRE that cover [OIII] in 12 quasars with 2.1<z<2.3 , selected to have strong outflow signatures in CIV {lambda} 1550. Combining with data from the literature, we build a sample of 73 BAL, 115 miniBAL, and 125 non-BAL quasars with 1.5<z<2.6. The strength and velocity width of [OIII] correlate strongly with the C IV emission properties, but no significant difference is seen in the [OIII] emission-line properties between the BALs, non-BALs, and miniBALs once the dependence on CIV emission is taken into account. A weak correlation is observed between the velocities of CIV BALs and [OIII] emission, which is accounted for by the fact that both outflow signatures correlate with the underlying CIV emission properties. Our results add to the growing evidence that BALs and non-BALs are drawn from the same parent population and are consistent with a scenario wherein BAL troughs are intermittent tracers of persistent quasar outflows, with a part of such outflow becoming optically thick along our line of sight for sporadic periods of time within which BALs are observed.