If you are the boss (root) change the password (and maybe upgrade the scripts calling or sending the encrypted old password if necessary). That's the short story

The long story is that I just realize that under some circumstances you could do what you want, tunning carefully sudoers (with visudo).

I merely think that's possible, but I'm never tested it and it depends on your case (ie: what is in your fstab) and also depends on if to add "your_other_user" to the group www-data could be a wise step or not in your scenery. Think about this carefully cause this could be a security hole

You probably don't want or need to mess with this, but it's another way. read man sudoers