I certainly wouldn’t throw the baby out with the bathwater or the crush equipment out with the sanitizing solution in this case. A little accumulated mold on a wooden basket press or fermentation barrel doesn’t faze this Wine Wizard. It shouldn’t phase you, either, because it sounds like you are taking all the appropriate steps to clean (scrubbing to remove the surface film and mold) and then sanitize (do a deeper dig with a potassium metabisulfite solution) your equipment.
Absolutely, mold spores can get into wood and are sometimes never totally removed. However, you have to look at how you use your equipment. Though I can’t make any guarantees against some heretofore unidentified voracious wine-ruining super-bug, I would wager that the short time the wine is in contact with your wooden basket press while pressing is not long enough to cause major contamination. The wine will be “living” in your fermentation vessel a little bit longer, of course, but during its time there presumably you have inoculated the must with a strong, reliable yeast strain (like I always suggest to my readers). During an active yeast fermentation, those organisms will dominate the environment (that is the idea for a dry, complete fermentation) and will usually out-compete the potential bad guys that might be lurking in your wood.
The cast iron parts should be no problem. If you’re worried, try cleaning using a caustic cleanser on the basket followed by a sulfite or citric acid soak.
Now if you had a storage barrel that was super-covered in rabid-looking fuzzy mold? That might make me pause a little bit. Finished wine doesn’t have the rampant microbial population of a young, strong fermentation and so is a little bit more fragile and needs our help defending itself. It also will be hanging out in that barrel for a long time (presumably about 12 months?) so its house needs to be clean to start with or all sorts of unwanted microbial houseguests could take root. Little bits of mold that easily come off, don’t come back and don’t cause any kind of lingering aromas don’t make me too worried to use a barrel again. But major mold infections in barrels that cause sour smells, volatile acidity or ethyl acetate aromas during empty storage, especially if the smell comes back after a good cleaning, should probably be turned into planters. I feel you’re 100% OK to use your basket press, but leave it to your judgment about how bad the mold in the wooden fermenter looked as to whether you want to use it again.