I had just gotten the car to start and stay running after replacing a number of parts including a proper tune up. I locked the car up, and upon returning to unlock the passenger door, the alarm went off. I tried disabling it using the driver side door, but alarm continued. I let it sit for a time, after the alarm stopped, and tried to lock and unlock both doors, then the trunk lid. The security light in the dash still flashes twice, pauses about 3 seconds and flashes twice repeatedly.

Is the problem with the alarm system or the power door locks? Any help would be appreciated.

 

"I locked the car up, and upon returning to unlock the passenger door, the alarm went off."
You didn't state how you unlocked the passenger door. If you reached in and popped the lock button from the inside, then the alarm did exactly what it's supposed to do. The only ways to open the car without having the alarm sound is do it with the key or remote.