...raw reads were mapped to the National Center for Biotechnology
Information (NCBI) build 37 using the Burrows-Wheeler Aligner (BWA)
“bwa mem” v.0.7.5a with default parameters
Variants were mapped to human reference coordinates (NCBI build 37) by
walking the read overlap graph in both directions until an “anchor”
read, where a continuous 65 bps matches the reference, denoted the
beginning and end of each variant.
It looks like it's GRCh37: On p. 11 of Parikh et al. (2014) in the first paragraph of "Methods", they say that they mapped the Platinum Genomes 2x100bp HiSeq data to NCBI "build 37" using bwa mem v.0.7.5a with default parameters, and that aligned (meaning, presumably, to the same reference) BAM files were publicly available for the Illumina 250bp, PacBio and Moleculo data. Also near the top of the next page they write that the Spiral Genetics variants in category C were mapped to NCBI build 37, though it's not yet clear to me how the subheading this falls under corresponds to the 11 rows of Table 2.