Living "A dual existence" could mean being someone at a particular moment and another person at another moment.
White mentioned being a father in the present and being a son in the remembered life of the past.
This means White was first a "<em>son</em>" to his "<em>father</em>" in the past and he is now a "<em>father</em>" to his "<em>son</em>" in the present. Therefore, he is living two different lives; life as a <em>son</em> and life as a <em>father</em>.