It appears from what they have written in the video that they do not use the epipen themselves when there is a reaction, but wait for the nurses/doctors at the ER to do so. I hope that it is just a misscommunication and they use the epi at first signs of a reaction. They are taking huge risks by not using the epipen right away and this delay in using the epi could result in tragic results.
It is also unclear if they are in a desensitization program. Other than a couple of shots at the beginning showing her holding peanut butter all the other photos are of her reactions.