You want 1 America the beautiful pass.
(I do not 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 carefully read the rules for this passage, it says:
Enables passport owners and accompanying passengers to enter nationwide recreational areas in a single private, non-commercial vehicle.
Covers the holder of the passport and three (3) adult escorts from the age of 16 in places where entrance fees are charged per person. No entry fee will be charged for children under 15 years.
The 4-person limit applies only to "locations where entry fees per person are charged". That does Not apply in the large national parks; They are all calculated per vehicle. For example, the Grand Canyon toll page displays a vehicle registration fee. This pass will allow you to enter all 5 of them, as they are all in a private, non-commercial vehicle.
(If you would enter the parks by bus, on foot or otherwise, there would be a limit of 4 people, and you'd better buy two passes.)
I do not think there is a shorter period than a year. Certain groups have very discounted or free tickets available, for example:
US citizens or permanent residents who are at least 62 years old or permanently disabled
Members of the US military, the reserves, the National Guard and their families
People who volunteered for a federal recreation area for at least 250 hours a year
US students of the fourth grade
So if someone in your group meets one of these criteria, they can get the pass. All these rules apply to everyone in the vehicle.
Otherwise you have to pay the full price. You can order the passport online, but you can buy it as soon as you enter your first park to avoid the $ 5 handling fee.
Finally, note that the Antelope Canyon is not a US federal city, but in the Navajo tribal land. Therefore, none of the above-mentioned passes grants you entry there. You have to pay separately. You can only drive there with an authorized tour. A prior reservation is required.