Although boron nitride nanotubes (BNNT) and hexagonal-BN (hBN) are superb one-dimensional (1D) and 2D thermal conductors respectively, bringing this quality into 3D remains elusive. Here, we focus on pillared boron nitride (PBN) as a class of 3D BN allotropes and demonstrate how the junctions, pillar length and pillar distance control phonon scattering in PBN and impart tailorable thermal conductivity in 3D. Using reverse nonequilibrium molecular dynamics simulations, our results indicate that although a clear phonon scattering at the junctions accounts for the lower thermal conductivity of PBN compared to its parent BNNT and hBN allotropes, it acts as an effective design tool and provides 3D thermo-mutable features that are absent in the parent structures. Propelled by the junction spacing, while one geometrical parameter, e.g., pillar length, controls the thermal transport along the out-of-plane direction of PBN, the other parameter, e.g., pillar distance, dictates the gross cross-sectional area, which is key for design of 3D thermal management systems. Furthermore, the junctions have a more pronounced effect in creating a Kapitza effect in the out-of-plane direction, due to the change in dimensionality of the phonon transport. This work is the first report on thermo-mutable properties of hybrid BN allotropes and can potentially impact thermal management of other hybrid 3D BN architectures.