Is a cop allowed to keep and use a police siren in their civilian car?

The other day I was practicing parkour in a ditch around my house, and this "cop" who was in a blue honda civic pulls up to me and uses a police siren. He drives up to me and says if he was on duty, he could arrest me for trespassing. Sure, I may have been trespassing, but is an off-duty cop allowed to have a siren in their non-cop car?

Depends on state and local law. For example: it's very common in Hawaii.

First, there are THREE types of vehicles an officer may drive.
1. A marked police vehicle.
2. An unmarked police vehicle. These usually have a siren, and in some cases are driven home by the officers they are assigned to.
3. A personally owned vehicle. This would be what you refer to as their 'civilian car'. It is usually up to the department to decide if sirens are permitted in POVs.

Also note: In nearly all US jurisdictions, an officer CAN make an arrest when 'off duty'. They normally don't want the hassle unless something serious is happening.