Does a Temporary Resident Permit let my family travel with me if they are not themselves inadmissible?

A TRP is issued to a specific individual for their own inadmissibility, so family members who are not themselves inadmissible generally don't need one to travel with the permit holder — they can typically travel and seek entry to Canada under their own status, such as a regular visitor visa, eTA, or other applicable authorization, applied for and assessed independently of the TRP holder's situation.

That said, "travelling together" and "being admitted together" aren't quite the same thing procedurally. Each family member is still individually assessed at the border or through their own application, even if the family is travelling as a unit and for a shared purpose, such as a wedding or family event. A non-inadmissible family member generally faces no special barrier connected to the TRP holder's situation, but they still need to meet the ordinary requirements for whatever status or document they're entering on.

Because the family's various documents and applications can be on different timelines and require different paperwork, it's worth planning the TRP application and the rest of the family's travel arrangements together and early, rather than assuming approval for one automatically clears the way for everyone else.