It sort of does and sort of doesn't. You can't just plop in the PAR or GS or whatever code into the BizHawk's cheats menu (or so I am led to believe) because the cheats menu is just a glorified Watch/Freeze/Poke interface - it works directly with the memory addresses, and PAR/GS/ETC codes have to include instructions and target values, not just the addresses themselves, so that the
physical device knows whether to freezepoke or to read from a different address or whatever. Not too sure about Game Genie but I'd presume it works similarly.
If you know how to extract the desired address from a cheat code, it's great. If not, I do remember seeing a hacking guide on Gameshark at some point that explained why most of those codes started with 8's even though that particular memory domain went unread most of the time. I took notes down from it in any case, just need to remember where I found it/where I put my notes.
E:
Found it. Or the Gameshark one, at any rate.