We present a class of parallel preconditioning strategies built on a multilevel block incomplete LU (ILU) factorization technique to solve large sparse linear systems on distributed memory parallel computers. The preconditioners are constructed by using the concept of block independent sets. Two algorithms for constructing block independent sets of a distributed sparse matrix are proposed. We compare a few implementations of the parallel multilevel ILU preconditioners with different block independent set algorithms and different coarse level solution strategies. We also use some diagonal threshholding and perturbation strategies to enhance factorization stability. Numerical experiments indicate that our parallel multilevel block ILU preconditioners are robust and efficient.