Should Parehanee tharpanam be performed the next day after performing a shrAddham? If so, is it done before or after a bath? Must one avoid partaking rice the night of parehanee tharpanam? Pray clarify what other restrictions must be observed.

Parehanee tharpanam is done on the day following a shrAddham as a component of the main shrAddham. Pare Ahanee means ‘during the following day’. Rules related to the cleanliness of Amavasya are applicable here.
Notes:
Parehanee tharpanam must not be performed before sunrise. It is performed after taking bath, adorning Thirumann-Srichoornam, performance of morning sandhyavandanam.
One must fast during the nighttime on the day of parehanee tharpanam. Those who are unable to fast can partake fruits. No other specific restrictions are applicable.