While there are military members on this forum, I would think that this question could more easily and definitively be answered by a recruiter. Might want to start there.
My guess: usually Army regs are in place to ensure that your health will not put others at risk or prevent you from being able to fight. Perhaps penicillin is one of the main antibiotics they utilize in the field, if you were to fall ill and require antibiotics but were unable to take penicillin due to an allergy, now you are a soldier that remains sick and unable to fight and will most likely have to be evacuated. There are multiple variations of this scenario that could cause problems. Makes sense to me.
As to the fact that you are only "mildly" allergic. They have to draw the line some where, its easier and less risky to deny any level of the allergy, rather than deal with the gray space of rashes and such.