When the FSX Sim starts up the plane will begin to roll. It doesn't matter which plane or where it's parked. I use a Saitek X52 yoke with the parking brake at the trigger. I'll sqeeze the trigger and the plane will stop but only for a few seconds. It's a nuisance when I'm doing the key board. I know the letter P will pause it, but that pauses everything. Is this something that everyone experiences  or is there an adjustment that can be made?

What plane? Is your throttle not idle? Happens at every airport? Does your parking brake just go on and off or do you see the message parking brake on your screen still?

Try changing your default flight:

1) Start as normal.

2) Stop the plane

3) Set parking brake

4) From the menu, select World | Go To Airport, and select your airport and gate of choice.

 

Hopefully, you'll find yourself parked still at that gate. If so, save this as your default flight, and that may do the trick

Are the engine indications/sounds suggesting that the throttles are commanding idle thrust?

Ensure that you close the hardware throttles at the end of each flight. I have found that FSX occasionally "remembers" the most recently used thrust setting and applies it at the beginning of the next flight. Also ensure that your throttle sensitivity is high enough so that the closed position of the hardware throttle commands idle thrust (vs. a bit of thrust).