We study the spin injection efficiency into single and bilayer graphene on the ferrimagnetic insulator Yttrium-Iron-Garnet (YIG) through an exfoliated tunnel barrier of bilayer hexagonal boron nitride (hBN). The contacts of two samples yield a resistance-area product between 5 and 30 k$\Omega\mu$m$^2$. Depending on an applied DC bias current, the magnitude of the non-local spin signal can be increased or suppressed below the noise level. The spin injection efficiency reaches values from -60% to +25%. The results are confirmed with both spin valve and spin precession measurements. The proximity induced exchange field is found in sample A to be (85 $\pm$ 30) mT and in sample B close to the detection limit. Our results show that the exceptional spin injection properties of bilayer hBN tunnel barriers reported by Gurram et al. are not limited to fully encapsulated graphene systems but are also valid in graphene/YIG devices. This further emphasizes the versatility of bilayer hBN as an efficient and reliable tunnel barrier for graphene spintronics.