What is the appropriate discretionary punishment (ta'zir) for someone who describes those who practice the Sunnah of letting the beard grow as belonging to backward political parties? And does this fall under the category of discretionary punishment for every sin for which there is neither a prescribed penalty (hadd) nor expiation (kaffarah)?

Ta'zeer is the legitimate punishment for every transgression for which there is no prescribed Hadd (fixed penalty) or Kaffarah (expiation). Its basis lies in the Qur'an, Sunnah, and scholarly consensus.

Ibn al-Qayyim and Sheikh al-Islam Ibn Taymiyyah hold that the determination of Ta'zeer is left to the ruler, based on what serves the public interest and the gravity of the crime.

Thus, if a person mocks any of the rituals of the religion, the determination of Ta'zeer falls to the ruler.

Your duty towards such individuals is to advise them with wisdom and remind them of Allah and the danger of their actions, as it might lead them to apostasy from Islam.

Sheikh Ibn Baz was asked about the ruling on mocking religion, and he replied that mocking religion is disbelief and apostasy, citing the verse: ﴿Say, "Is it Allah and His verses and His Messenger that you were mocking?" Make no excuse; you have disbelieved after your belief.﴾ [9:65-66]. However, if the mockery is directed at the person himself for other reasons, and not intended as mockery of the religion, then it is not disbelief, but it is a reprehensible act, and its perpetrator is in danger.