Which is better after Hajj or Umrah: visiting the Prophet’s Mosque (peace be upon him) or visiting his grave? And which is more prioritizing for one who can travel after a long absence: performing Hajj for the first time or visiting his elderly parents?

The Sunnah is to visit the Prophet's Mosque, not the noble grave. And setting out on a journey is only for the three mosques: al-Masjid al-Haram, al-Masjid al-Aqsa, and al-Masjid an-Nabawi. Visiting the grave comes as a consequence of visiting the Mosque, and is not intended in itself when setting out on a journey. Traveling to visit the graves of prophets and righteous people is an innovation (bid'ah) that none of the Companions or their successors (Tabi'in) ever did.

As for Hajj, it is obligatory immediately when its conditions are met, and this is the preferred view among scholars. The evidence for this is the saying of the Prophet (peace be upon him): "Whoever intends to perform Hajj, let him hasten, for indeed the sick person may fall ill, the ride may go astray, and a need may arise." Therefore, Hajj must be performed first, while seeking the parents' pleasure and traveling to them whenever it is possible.