I have a Goodman furnace about 15 yrs old. Problem is it starts up by itself with the thermostat off. I replaced the thermostat but it still does the same thing.

Sounds like you have a short circuit in the low voltage wiring from the thermostat to the unit.

Is it gas or electric furnace?

If its electric after you`ve check the low voltage wiring for short,, it just might be the sequenser

They never give us enough information and my crystal ball is still broken waiting for parts.

Are we talking fan or heat and fan?

We are talking heat and fan with the thermostat set to the lowest setting. I also completely removed the thermostat and kept the wires separated, but when I switched on the emergency switch the heater started up. (I turned off the emergency switch so it doesn't start up) It is Goodman a gas furnace.