We could have just as easily said family, but "Persephone, Falling" is really all about mothers and daughters. Dove is trying to reconcile all the thoughts and desires she had when she was a child with all the fears and anxieties she has now that she is a mom. In the poem, Persephone is a daughter longing to be free of her mother's overprotection, while her mother longs to keep her safe, innocent and childlike.
Persephone's desire to distinguish herself from her mother chases her right into the arms of a dangerous man.
Demeter's real struggle in this poem is not how to keep her daughter safe, but how to let her go.