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