Is it permissible to perform Istikhara prayer for a trip decided by the parents and committed to by the family, and is the Istikhara of one person sufficient for everyone?

Istikhara (guidance-seeking prayer) is for permissible matters, not for obligatory or forbidden ones. If the journey is forbidden, then Istikhara is not prescribed, and obedience to parents is not obligatory in it. If it is permissible, and your parents insist that you travel with them, then obedience to them is obligatory, and Istikhara is not prescribed for you, but rather for the parents. However, if the parents do not insist on the journey, then Istikhara is prescribed for every individual in the family; because each person has a choice in their travel, and deputizing someone for Istikhara is a matter of scholarly disagreement.