Phase transitions, compensation phenomenon, and magnetization of a ferroferrimagnetic ternary alloy AB_{ρ}C_{1-ρ} composed of three different kinds of magnetic ions A, B, and C with the spin magnitudes 1/2, 1, and 3/2 are examined within the framework of a mixed-spin Ising model on a honeycomb lattice with a selective annealed site disorder on one of its two sublattices. It is supposed that the first sublattice of a bipartite honeycomb lattice is formed by the spin-1/2 magnetic ions, while the sites of the second sublattice are randomly occupied either by the spin-1 magnetic ions with a probability ρ or the spin-3/2 magnetic ions with a probability 1-ρ, both being subject to a uniaxial single-ion anisotropy. The model under investigation can be exactly mapped into an effective spin-1/2 Ising model on a triangular lattice through the generalized star-triangle transformation. For a specific concentration of the spin-1 (spin-3/2) magnetic ions, it is shown that the ferroferrimagnetic version of the studied model may display a compensation temperature at which the total magnetization vanishes below a critical temperature. The critical temperature strikingly may also become independent of the concentration of the randomly mixed spin-1 and spin-3/2 magnetic ions for a specific value of a uniaxial single-ion anisotropy. The spontaneous magnetic order may be notably restored at finite temperatures through the order-by-disorder mechanism above a disordered ground state, which results in an anomalous temperature dependence of the total magnetization with double reentrant phase transitions.