Is saying "Lā ʾilāāha ʾillā-llāh" (There is no god but Allah) and then responding with "Muḥammad rasūlullāh" (Muhammad is the Messenger of Allah) upon parting a Sunnah or an innovation?

There is no authentic (sahih) or weak (da'if) hadith that specifies a particular dhikr (remembrance) for parting or concluding a gathering, other than the well-established supplications (ad'iya ma'thura). Persisting in an unprescribed dhikr or believing it to be a Sunnah constitutes an innovation (bid'ah). The fundamental principle regarding acts of worship (ibadat) is that they are based on divine injunction (tawqif 'ala al-shar').

It is prescribed to say at the conclusion of a gathering: "Subhanak Allahumma wa bihamdika, ashhadu an la ilaha illa Anta, astaghfiruka wa atubu ilayk" (Glorified art Thou, O Allah, and praised be Thou, I bear witness that there is no god but Thou, I seek Your forgiveness and repent to You). This serves as an expiation for what may have occurred in the gathering. It is also prescribed for two people who meet to recite Surat Al-Asr before parting.

People may abandon established Sunnah practices for the sake of innovations, in accordance with the Prophet's (peace and blessings be upon him) saying: "No people introduce an innovation but that a similar amount of Sunnah is removed from them."