It's actually very easy to do that.
Just buy another phone number, design your own circuit board to put your phone card in it, and connect the board to your computer, and program some software so your computer will execute a certain command.
Your computer can do pretty much anything, if you know how to program the idea you have.
I don't know if such circuit boards are on the market, but they aren't that hard to make...
Just imagine a regular smartphone, now in your imagination remove the Screen, Speakers, Buttons, Camera, Battery, All the connection ports, Storage devices (MicroSD and internal memory), the Frame, Microphone.
And you leave a simple circuit board of 5 cm² (3.14 square inch) perhaps a little bigger.
I know this is a pretty easy and non-professional way, but this is how the pro's do it too, but on a larger scale.