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 enter Federally operated recreation sites across the country.
Covers the pass owner and three (3) accompanying adults age 16 and older at sites where per person entrance fees are charged. No entry fee charged for children 15 and under.
The 4-person limit only applies at "sites where per person entrance fees are charged". That does not apply at the major national parks; they are all charged per vehicle. For example, the fees page for Grand Canyon shows a vehicle entrance fee. So this pass will let all 5 of you enter, since you will all be in a single private non-commercial vehicle.
(If you were going to enter the parks by bus or on foot or some other way, then the 4-person limit would apply, and your best bet would be to buy two passes.)
I do not believe there is any pass for a shorter period than one year. There are deeply discounted or free passes available to certain groups, such as:
US citizens or permanent residents who are at least 62 years old or permanently disabled
Members of the US military, reserves, National Guard, and their dependents
People who have volunteered at least 250 hours in a year at a federal recreation site
US fourth grade students
So if anyone in your party meets any of those criteria, you could have them get the pass. All of these have the same rules covering everyone in the vehicle.
Otherwise, plan on paying the full price. You can order the pass online, but you can also just buy it when you enter your first park, avoiding the $5 handling fee.
Finally, note that Antelope Canyon is not a US federal site, but on Navajo tribal land, so none of the aforementioned passes will grant you admission there. You will have to pay for it separately. You can only go there with an authorized tour and advance reservations are required.