Of course you know you can't move the shifter in or out of park unless your foot is on the brake pedal, so I won't mention that.
I think a 2001 is old enough to still have a hard wired brake light button switch on the brake pedal. The problem is probably the switch. It detects whether the brake is depressed or not. See if you can find a plunger type switch up at the top of your brake pedal arm, and check to see if it is mounted securely and adjusted so the brake lights come on just as the pedal starts moving.
Check also to see if your brake lights are coming on, they may or may not be working, or they may work sometimes.
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Bone Stock '07 GT 4.6, Redfire w/ parchment leather, Automatic.
If it looks right, it is right, and I decide if it looks right. Paraphrasing Sir Sydney Camm (of Hawker aircraft IIRC)