I am writing a story and one of the characters is a paramedic.?

Yes paramedics have supervisors.

A paramedic can work as part of a fire department or might drive an ambulance for a hospital. This means different unions usually. If they're part of a fire department, they're under the department hierarchy and work similar shifts. If it is a hospital, then it'll depend on the size of the hospital and how many people are on duty.

A small town fire department operates differently than a big one. The same goes for hospitals.

If you really want to do a good job at this, I suggest finding a paramedic in a similar situation and interview him, tell him you're writing a book or that you're interested in a such a career.

That is how you get details right.

Surely this depends on the employer, country, etc?

In the UK, we have a national health service (tax payer funded). Healthcare is divided down to a regional level (equivalent to states in federal countries, but we're not one) and each NHS trust has their own hospitals, which potentially could all have their own policies.

In the US, healthcare is privatised, and so I imagine rules regarding working hours etc is done on a per-company/voluntary organisation basis, with each provider having their own rules/regs.

As James has pointed out, requirements of your plot dictate this as much as anything. If it's set in a particular town, you could even chase it up yourself - the organisation might even publish sample employment contracts. Google is your friend!

The answer depends on the service. As long as you stay consistent you can give your medic any working conditions/schedule that you need for the requirements of your plot.