Bhikhshuvar (monk/mendicant) + Adikal (reverential addressing / honorific suffix) = Bhikhshuvaradikal, later corrupted to Pisharodikal and pisharody ( Shara itself being a term for a younger sanyasi).
According to this, Pisharodys are descendants of a group of highly respected pure brahmans who intended to be Sanyasis.
By then they had discarded almost all basic rituals usually performed by an ordinary brahman being elevated to the next level and hence they could not return to their earlier sect.
Since they are already elevated to Sanyasi thought process, they continued as advisors to temples and authentic teachers of scriptures and higher level sanskrit.
Pisharodys do not need to wear the sacred thread nor perform puja in temples as per the tradition, but act only as advisors.