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