Does my Mac need to be awake, or my iPhone plugged in?

On Mac

Yes, your Mac needs to be awake and Parachute Backup needs to be running at the scheduled time. It is an app, not a system service. Once a backup has started, Parachute Backup keeps your Mac from going to sleep until it is done, and the display can turn off.

For backups at night, turn on Start at Login and keep your Mac from sleeping during that time.

On iPhone and iPad

For scheduled backups, yes. iOS decides when apps may work in the background, and Parachute Backup asks to run while your device is charging and online. Also make sure that Background App Refresh is on for Parachute Backup, Low Power Mode is off, and notifications are allowed with Immediate Delivery.

If iOS stops a run early, nothing is lost: the next run continues with what is missing.