The hiring mgr is usually the person you will report to, or is in charge of the team (ie theres another manager who will supervise you day to day). S/he’s the guy you need to impress.
If they explicitly say no calls, don’t call. There’s no faster way to piss someone off then to do something they said not to do. They even went out of their way to write that in the ad!
If you insist on reaching out, try to find a person in the org who you can network with and learn about the role etc. If you hit it off, he may help your resume along. If not, they won’t. If you haven’t screwed up that convo, then email the hiring manager with a quick blurb showing you know what the job is about. Keep it brief, make it easy to read.