When I had Bell's Palsy the first time, about 25 years ago, there was a simple and definitive explanation from my doctor.  It is caused by an inflammation of the mastoid sinus (slightly behind and below the ear on either side, rarely on both sides which results in the atypical effect of temporary paralysis of both sides of the face).  The inflammation is usually caused by an infection, and typically folllows a cold or a canker sore in the mouth.  The inflammation of the sinus causes it to swell, which exerts pressure on the trigeminal nerve, a main nerve controlling most of the muscles of the face roughly as far as the middle of the face laterally but not the deeper muscles in the facial area (NOT the tongue or jaw muscles, but INCLUDING the LIPS, or at least the fine motor control of the lips [if you try to drink from a cup you will almost certainly "drool" out of the paralyzed side of the lips, usually you have to drink with a straw].  This loss of control of muscles of the face usually also affects the eyelid (on whichever side the trigeminal nerve is being "pinched").  It is advisable to take great care with the eye since it does not close firmly when sleeping and can easily scratch the cornea on blankets or sheets, or allow dust or even bugs or larger bits of dirt into the eye.
Now, when I had Bell's Palsy more recently, the sweet young doctor thought she had to do several "tests" before she would "concur with" my self-diagnosis, but she also told me that there are now other "theories" about "causes" of Bell's Palsy, though she was not very specific about what the other theories were.  
Bell's Palsy is usually a temporary condition that lasts anything from a couple of weeks up to 6 months or (rarely) even a year.  Very rarely it can become permanent, especially if not treated promptly, but since these are usually viral infections doctors are sometimes reluctant to use antiviral medicines.  Incomplete or ineffective treatments with antivirals actually promotes "stronger" virii, more resistant to the medicines that would normally defeat them.
Now, having put you through all that "education" about Bell's Palsy, it is unlikely that an allergic reaction to an antibiotic medicine would cause the lining of the mastoid sinus to become inflamed.  However, it is certainly possible (especially if she stopped taking the medicine before the pills ran out or otherwise using all that the doctor had prescribed) that whatever infection was being treated with the antibiotic could have spread to the mastoid sinus causing inflammation that resulted in pressure on the trigeminal nerve and thus the palsy.
I am not a doctor, and this information should not be used in substitution for consulting a medical profession for any healthcare matter.
My "nickname" is "Doc" but that is only because I tend to be interested in and remember information like the details above.  I often make mistakes, however, and you DEFINITELY want to get professional medical advice for any medical problem.
 
Sincerely,
Stafford "Doc" Williamson