The phase behavior of suspensions of colloidal hard tetragonal parallelepipeds ("TPs") (also known as rectangular nanorods or nanobars) was studied by using Monte Carlo simulations to gain a detailed understanding of the effect of flat-faceted particles on inducing regular local packing and long range structural order. A TP particle has orthogonal sides with lengths a, b, and c, such that a=b and its aspect ratio is r=c/a. The phase diagram for such perfect TPs was mapped out for particle aspect ratios ranging from 0.125 to 5.0. Equation of state curves, order parameters, particle distribution functions, and snapshots were used to analyze the resulting phases. Given the athermal nature of the systems studied, it is the interplay of purely entropic forces that drives phase transitions amongst the structures observed that include crystal, columnar, smectic, parquet, and isotropic phases. In the parquet phase that occurs for 0.54<r< or =3.2, for example, the particles possess some translational entropy (mobility) but reduced orientational entropy; particles arrange in stacks oriented perpendicular to one another, so that all particle axes are aligned along three common directors. Multicanonical-type simulations were used to study in more detail the isotropic-parquet phase transition. Both similarities and differences were identified between the results for these perfect TPs and those unveiled in our previous study of approximate (polybead) TPs.