Recently, parity-declustered layouts have been studied as a tool for reducing the time needed to reconstruct a failed disk in a disk array [5, 9]. Construction of such layouts for large disk arrays generally involves the use of a balanced incom- plete block design (BIBD), a type of subset system over the set of disks. This research has been somewhat hampered by the dearth of general results and constructions for BIBDs on large sets, and by inefficiencies in some parity-distribution methods that create layouts that are larger than necessary. We make progress on these problems in several ways. In particular, we • Demonstrate a new BIBD construction that general- izes some previous constructions and yields a simpler BIBD that is optimally small in some cases. Show how relaxing some of the balance constraints on data layouts leads to constructions of approximately- balanced layouts that greatly increase the number of feasible layouts for large arrays. • Give a new method for distributing parity that pro- duces smaller data layouts, resulting in tight bounds on the size of data layouts derived from BIBDs. Our results use a variety of algebraic, combinatorial, and graph-theoretic techniques, and together greatly increase the number of parity-declustered data layouts that are ap- propriate for use in large disk arrays.