A person cannot perform Hajj on behalf of more than one person in a single Hajj season. Hajj is a complete act of worship with a unified intention (niyyah), and each performance can only be designated for one person. Just as a person cannot perform their own obligatory Hajj and simultaneously fulfill it for someone else, they cannot combine two proxy Hajjs into one performance.
Sheikh Ibn Baz stated clearly that each Hajj season, a proxy can perform Hajj for only one person. If someone wants to perform Hajj for their deceased father and mother, they would need to perform Hajj for one of them this year and for the other in a subsequent year. He advised prioritizing the parent who passed away first or the one with the greater right.
Sheikh Ibn Uthaymeen confirmed this ruling and added that some scholars discussed whether the general reward of a voluntary Hajj can be shared among multiple deceased through supplication and intention of gifting the reward. However, the actual Hajj rites — the ihram, intention, talbiyah — can only be for one specific person. He distinguished between the formal proxy Hajj (which is specifically for one person) and general supplication for the deceased (which can include multiple people). A person performing Hajj may make dua for as many people as they wish, but the formal Hajj itself is for one individual only.