We consider deterministic dictionaries in the parallel disk model, motivated by applications such as file systems. Our main results show that if the number of disks is moderately large (at least logarithmic in the size of the universe from which keys come), performance similar to the expected performance of randomized dictionaries can be achieved. Thus, we may avoid randomization by extending parallelism. We give several algorithms with different performance tradeoffs. One of our main tools is a deterministic load balancing scheme based on expander graphs, that may be of independent interest. Our algorithms assume access to certain expander graphs "for free". While current explicit constructions of expander graphs have suboptimal parameters, we show how to get near-optimal expanders for the case where the amount of data is polynomially related to the size of internal memory.