If you look at the Task information:
hybernate = qmsuspend
pause = qmpause
API docs state:
suspend: You need 'VM.PowerMgmt' on /vms/{vmid}, and if you have set 'todisk', you need also 'VM.Config.Disk' on /vms/{vmid} and 'Datastore.AllocateSpace' on the storage for the vmstate.
The same /suspend API call is used, the difference is in the parameters specified