Allergies can be manifested as a cough. What is really going on is similar to a child with asthma and over time it may very well turn out that the child has asthma and that certain things trigger it such as pet dander - ie child isn't allergic to dogs in the true sense just reacts to them if that makes sense. When the child starts coughing they get their inhaler to open the airways and they do better. In the absence of any other symptoms of a cold (fever, runny nose, etc.) it is treated as asthma.

As to a doctor's note there is a cost for that in most cases and parents are not going to run to the doctor at each symptom of anything - they are going to do exactly what this parent did - send the child to a relative or take the day off to see if a cold actually develops. If it does they stay home, if child gets no worse they go to daycare. As to the lying - that is a result of setting the parent up to have no choice because they were doing a wait and see approach which is totally acceptable in this case but they need to tell you what you want to hear so child that just has a cough from 9 I would say probably asthma over allergies) can attend daycare.