An all-optical centimeter-scale laser-plasma positron accelerator is modeled to produce quasimonoenergetic beams with tunable ultrarelativistic energies. A new principle elucidated here describes the trapping of divergent positrons that are part of a laser-driven electromagnetic particle-shower with a large energy spread and their acceleration into a quasimonoenergetic positron beam in a laser-driven plasma wave. Proof of this principle using analysis and particle-in-cell simulations demonstrates that, under limits defined here, existing lasers can accelerate hundreds of MeV pC quasi-monoenergetic positron bunches. By providing an affordable alternative to kilometer-scale radio-frequency accelerators, this compact positron accelerator opens up new avenues of research.