Paramedics can be called within the Emergency menu. Unlike doctors, which should stay in the Infirmary, paramedics can heal injured prisoners and staff on-the-fly.
Paramedics can be used to heal Riot Guards who are dealing with a Riot and staff and prisoners who were injured in said event, as normal staff will not enter rioting zones or such with fire, while the paramedics won't care too much.
Paramedics are, like other emergency staff, controlled by the player. They are generally used when someone is bleeding out or when the prison is short of doctors.
Max. 2 teams of paramedics can be on site at once.
Paramedics have a first aid kit on their back suspiciously familiar to the ones from Valve's Left 4 Dead series of games.
The Paramedics used to have a red cross on their back, just like there was a red cross on their vehicle, the ambulance, as well as on the hearse. Both red crosses were later exchanged by green crosses on request of the Red Cross organisation. Introversion Software released a whole YouTube video about this and similar issues: "We broke the geneva convention".