They only pay for your parking at your base.
Not sure about DFW but if that's not your base some major airports don't give you access to employee parking.
You'll need to buy a monthly pass to your local park and fly place.
Most have discounts for airline employees.

That's the 121 world.
Buddy at American has to park at off airport parking at the airport he commutes out of also.
You don't need anything from the CP office for that.