A person cannot perform حج on behalf of more than one person in a single حج season. حج 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 واجب حج and simultaneously fulfill it for someone else, they cannot combine two proxy Hajjs into one performance.
Sheikh Ibn Baz stated clearly that each حج season, a proxy can perform حج for only one person. If someone wants to perform حج for their deceased father and mother, they would need to perform حج 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 حج can be shared among multiple deceased through دعا and intention of gifting the reward. However, the actual حج rites — the احرام, intention, talbiyah — can only be for one specific person. He distinguished between the formal proxy حج (which is specifically for one person) and general دعا for the deceased (which can include multiple people). A person performing حج may make دعا for as many people as they wish, but the formal حج itself is for one individual only.