Woah woah, hold on now before you go taking stuff out of your shifter. I had the same problem a couple years back with a different car, so here's the checklist that you should run through:
1.) Make sure you fully and completely compress the break pedal when you try to shift. It's more common than you think that that's the only problem!
2.) Are your brake lamps lighting up when you press on the brakes?
If not, then it's either a blown fuse (check number 33 and 41), or your brake lamp switch is malfunctioning. I can't remember off the top of my head how to check the switch, but I'm sure you can find that online. If your brake lamps are lighting up, then...
3.) The brake-shift interlock system has probably malfunctioned. This one is pretty common in a lot of vehicles. There should be some kind of manual override for this in the car somewhere, but I've never had to release one in a mustang so I wouldn't know (if anyone could help fill in the gaps that'd be great).
Past that, you might want to take it on in. If it isn't any one of those, then you probably have some kind of broken part that you'll need a replacement for. Anyways, hope I could help! Best of luck, man.