Unfortunately, you can't demand that a parent administers Benadryl since it isn't labeled for use in a child that young without a doctor's supervision. I say that mainly because you don't want to get yourself in trouble if you were to demand it and the child were to have some sort of reaction. As a daycare provider, you can't administer against labeling without a doctor's prescription.
Even if a child isn't "sick" snot contains all sorts of nastiness. However, a child this age can certainly be taught to help contain their own cooties (i.e. covering their mouth and nose). There is a chance that she is having an adverse reaction to something in your home. She actually MAY NOT be sneezing that much at her own home.
My daughter had a perpetually runny nose (along with many of her classmates) and the daycare repeatedly denied any culpability and refused to investigate or make any modifications. Within 4 days of changing daycares, her runny nose dried up and she has been sick only once since April--and NOT with a cold/runny nose.
If you think this child needs to go to the doctor, by all means notify the parent and give a firm deadline for resolution. If they continue to refuse, then you really have no choice but to refuse them care. Just be as understanding as you can about them needing to find another facility.