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Nonreciprocal elements, such as isolators and circulators, play an important role in classical and quantum information processing. Recently, strong nonreciprocal effects have been experimentally demonstrated in cavity optomechanical systems. In these approaches, the bandwidth of the nonreciprocal photon transmission is limited by the mechanical resonator linewidth, which is arguably much smaller than the linewidths of the cavity modes in most electromechanical or optomechanical devices. In this work, we demonstrate broadband nonreciprocal photon transmission in the \emph{reversed-dissipation} regime, where the mechanical mode with a large decay rate can be adiabatically eliminated while mediating anti-$\mathcal{PT}$-symmetric dissipative coupling with two kinds of phase factors. Adjusting the relative phases allows the observation of \emph{periodic} Riemann-sheet structures with distributed exceptional points (Eps). At the Eps, destructive quantum interference breaks both the $\mathcal{T}$- and $\mathcal{P}$-inversion symmetry, resulting in unidirectional and chiral photon transmissions. In the reversed-dissipation regime, the nonreciprocal bandwidth is no longer limited by the mechanical mode linewidth but is improved to the linewidth of the cavity resonance. Furthermore, we find that the direction of the unidirectional and chiral energy transfer could be reversed by changing the \emph{parity} of the Eps. Extending non-Hermitian couplings to a three-cavity model, the broken anti-$\mathcal{PT}$-symmetry allows us to observe high-order Eps, at which a parity-dependent chiral circulator is demonstrated. The driving-phase controlled periodical Riemann sheets allow observation of the parity-dependent unidirectional and chiral energy transfer and thus provide a useful cell for building up nonreciprocal array and realizing topological, e.g., isolators, circulators, or amplifiers.