Use of mold (of various sources, including bread and spider webs) stretches back into Antiquity, and may well be prehistoric.
The problem with that is that like most pre-modern medicine, it was treated as more an art than a science, and so there was no rigorous effort to isolate which kind of mold it was, how to grow it successfully, etc (I'm not even sure how many cultures would have realized that there's more than one kind of mold).
Developing penicillin as anything more than a particularly effective folk/traditional/alchemical/whatever medicine is that it requires three things:
-An understand of germ theory (that disease is caused by little animals, i.e. removing the animals can stop the disease)
-An ability to culture molds effectively
-Ideally, a formalized scientific method to isolate the correct mold.
Germ theory was proposed in the mid-16th Century and enjoyed a brief revival in the mid-18th century, but ultimately was not universally well-regarded until Pasteur in the mid-19th.
The formal cultivation of samples (including molds) OTL became widespread in England around the mid 19th century.
Most people like to say that the scientific method
per se comes with Sir Francis Bacon in the late 16th/early 17th centuries (though the "scientification of medicine" wouldn't happen until well into the 19th).
So - we can push penicillin up through the late 19th century with absolutely no problems. Earlier than that? OTL, maybe a century or two earlier. The big bottleneck in my opinion is germ theory, since cultures stem from that fairly easily (though, I mean, cell culture isn't a very technically vexing technology - if you skimmed some milk, then boiled it, then made sure to keep the result clean and covered, that's not crazy far from a modern petri dish).
And there...well, aside from a few references of Galen to "fever spores", I don't think the Romans had anything germ theory-ish. OTL germ theory was the result of a stream of studies that ultimately began from the microscope, though (there's tiny animals, oh god, there's tiny animals everywhere, wait, there appears to be some sort of mechanism to transmit disease between people close to each other, maybe the diseases are transmitted by these tiny animals?)
So...I guess, make microscopes.
Romans could make clear glass, and there was certainly enough knowledge of lenses and optics to figure it out. OTL, microscopes followed from high-quality telescopes, which I think were driven by sea exploration (both for navigation by precise sightings of the stars, and for looking at land to find landmarks etc).
I mean, you don't
need any of that to realize that mold sometimes stops wounds from festering, but you need all that in order to make it into a consistent, reliable treatment.
Also, do note that unlike vaccination which are done once for each person and require a needle and some pus, treating people with penicillin in this manner would require large amounts of cultured mold. Culturing the mold and keeping it uncontaminated is going to be...non-trivial before modern antiseptic techniques. This treatment is definitely going to be only for the wealthy, and may not even work all the time if culturing techniques aren't sophisticated enough to make sure the cultures are always pure and never contaminated with something nasty (or just something that looks like but isn't penicillin mold). Plus, not every bacteria responds to penicillin, and your medieval physicks aren't exactly going to know ahead of time which will and won't respond to treatment (hell, even modern doctors have trouble with this).
Also, let me be the first in this thread to remind you that "dark ages" is a major misnomer and that the period actually saw a great deal of development of every kind