First of all, the PSIO fits in the parallel port
Even then, it's not an actual parallel port accessory (which could autoboot just fine, if it was), but an ODE - it really needs that internal card to disconnect the optical drive and replace its signal with that coming from the parallel port (it's like those pci expess extension cables where one of the parts is an USB 3 cable, chosen because it's a common and affordable 9 pins connector, not because it has anything to do with USB)
To answer the literal question, you can run software from the serial port (that's how the Yaroze works, apart from the gratuitous added DRM to the official boot disc to make it incompatible with regular PS1s) but not only it's unsuitable for commercial games, you would need a boot disc (or ODE) in the first place
There can't be an equivalent of FMCB, because the official firmware doesn't run programs from memory card (again, boot disc, boot-parallel-card, or rom replacement required) and up to 2x128 KB are even more limiting than the almost 2 MB you get for running homebrew from serial