There are jobs for EMTs in St. Louis without being a firefighter, but it is not a great city for it, being that most the 911 services are fire based.
As an EMT in St. Louis you could apply to one of the private ambulance companies: Abbott, Gateway, or Christian. At Abbott they will hire nearly anyone with an EMT license that can pass the entry exam, and it's not difficult at all. I'm not sure about the other two companies hiring processes. Many EMS personnel in the area will work for one of the three companies at some point in their career. You will do a LOT of interfacility transfers at these companies, and very few 911 calls. Although it's decent experience when you're fresh out of school.
Another option if you are over 21 years old would be to work at one of the Casino's in the area as EMT/Security. You will get paid better at the Casino's than you would at a private ambulance company, but it's more security work and less EMS work.
On the Illinois there is MedStar, another private ambulance company, I've heard they pay better than the St. Louis companies, but I don't much about them.
St. Charles County Ambulance District (SCCAD) is the place to be if you want to make a career out of EMS in St. Louis without being a firefighter. But getting hired on there is not easy, if that's somewhere you want to work it would be in your best interest to go through their medic program.
What your dad is referring to is the St. Louis County Fire Academy. To get into the academy you have to enter into the Academy lottery or be sponsored by a department. Most St. Louis County Departments require you to put yourself through the academy, along with already being a medic, before being able to apply to their department. Although there are exceptions to this. Some departments in the area require no experience/training to apply to them, although preference may be given to those that have experience/training. St. Louis City Department for example does not require any experience or training to apply and gives no preference to those with it.
Also, just something to think about, you say you don't want to fight fires and want to be a medic, well firefighter's rarely fight fires, most of what they do is EMS work. And in St. Louis if you want to work the 911 calls, you need to be on a fire department, the private companies get some 911 calls, but nothing like the fire departments do.
If you have any more questions let me know.