Recently Rubinfeld et al. (ICS 2011, pp. 223--238) proposed a new model of sublinear algorithms called \emph{local computation algorithms}. In this model, a computation problem $F$ may have more than one legal solution and each of them consists of many bits. The local computation algorithm for $F$ should answer in an online fashion, for any index $i$, the $i^{\mathrm{th}}$ bit of some legal solution of $F$. Further, all the answers given by the algorithm should be consistent with at least one solution of $F$. In this work, we continue the study of local computation algorithms. In particular, we develop a technique which under certain conditions can be applied to construct local computation algorithms that run not only in polylogarithmic time but also in polylogarithmic \emph{space}. Moreover, these local computation algorithms are easily parallelizable and can answer all parallel queries consistently. Our main technical tools are pseudorandom numbers with bounded independence and the theory of branching processes.