  Phase spaces as given by the Wigner distribution function provide a natural
description of infinite-dimensional quantum systems. They are an important tool
in quantum optics and have been widely applied in the context of time-frequency
analysis and pseudo-differential operators. Phase-space distribution functions
are usually specified via integral transformations or convolutions which can be
averted and subsumed by (displaced) parity operators proposed in this work.
Building on earlier work for Wigner distribution functions [A. Grossmann, Comm.
Math. Phys. 48(3), 191 (1976)], parity operators give rise to a general class
of distribution functions in the form of quantum-mechanical expectation values.
This enables us to precisely characterize the mathematical existence of general
phase-space distribution functions. We then relate these distribution functions
to the so-called Cohen class [L. Cohen, J. Math. Phys. 7(5), 781 (1966)] and
recover various quantization schemes and distribution functions from the
literature. The parity-operator approach is also applied to the Born-Jordan
distribution which originates from the Born-Jordan quantization [M. Born, P.
Jordan, Z. Phys. 34(1), 858 (1925)]. The corresponding parity operator is
written as a weighted average of both displacements and squeezing operators and
we determine its generalized spectral decomposition. This leads to an efficient
computation of the Born-Jordan parity operator in the number-state basis and
example quantum states reveal unique features of the Born-Jordan distribution.
