You want 1 America the Beautiful Pass.
(I don't think there is any such thing as a "US Parks Pass". Your link, usparkpass.com, is just a third-party site selling the same America the Beautiful pass.)
If you read the rules for that pass carefully, it says:
Allows pass owner and accompanying passengers in a single, private, non-commercial vehicle to ...
The statement that the cited website is the official website for Antelope Canyon tours is false. I found a list of companies which offer tours to Antelope Canyon (note that the prices listed in the table are not accurate as of today). It looks like one can almost always find availability for any date since there are more than 10 companies organizing tours.
There are multiple Antilope Canyons you can visit.
Upper Antilope Canyon: https://navajotours.com
Lower Antilope Canyon: https://lowerantelope.com
Antelope Canyon X: https://www.antelopecanyon-x.com
During the summer the more popular are very crowded. (There are some nice videos on YouTube showing how uncomfortable it is for the visitors).