  We present a generic system of three harmonic modes coupled parametrically
with a time-varying coupling modulated by a combination of two pump harmonics,
and show how this system provides the minimal platform to realize nonreciprocal
couplings that can lead to gainless photon circulation, and phase-preserving or
phase-sensitive directional amplification. Explicit frequency-dependent
calculations within this minimal paradigm highlight the separation of
amplification and directionality bandwidths, universal in such schemes. We also
study the influence of counter-rotating interactions that can adversely affect
directionality and associated bandwidth; we find that these effects can be
mitigated by suitably designing the properties of the auxiliary mode that plays
the role of an engineered reservoir to the amplification mode space.
