Do I need both the Mac app and the iPhone/iPad app?

Usually not. When using iCloud Photos and iCloud Drive, most of the data is available everywhere. Parachute Backup for Mac can copy everything and a Mac is easier to keep awake, connected to power and an external drive for a potentially long-running initial backup. Parachute Backup Mobile is the right choice if you do not have a Mac, if you are away from it for longer stretches, or if you have files that only live on your iPhone or iPad.

For historic reasons, the two apps are separate purchases.