rear and side camera problem

Update, Today I tried running the monitor on the ACC and I doesn't work on the position on the ignition. It must have been a fluke where it decided to work for a bit. One good thing is the 3 camera's are working. I think I might replace the monitor.:(

Are the cameras wired or wireless?

I had this problem with my four cameras. You do not say whether your cameras are wireless or hard wired. Wireless are prone to interference than hard wired. . .although they are not totally immune to the problem you have.
When you are not running your engine you are drawing power from the battery which is clean DC.
When you run your engine, you are charging the battery from the alternator which puts out a bit higher than the battery voltage and a bit "dirtier". Installing a filter that feeds the camera can (sometimes) cure this.
If it doesn't, check the voltage to cameras (with engine running) to see if you are getting 12.7V. It should be no higher.

These made in China goodies are mostly designed for surveillance videos and work great with power supplies with stellar filtering systems.

tinkerer wrote:
I forgot to mention when the monitor is working it gives good pictures from all three sides.

Then I would think that you may have a ground problem at the monitor. This can be checked with a short jumper wire from a "ground" area on the monitor to any place on the chassis that is bare metal and attached to the main frame. Install the jumper ground wire and then start the engine to see if this corrects the issue.

I forgot to mention when the monitor is working it gives good pictures from all three sides.

Certainly would check ALL of your connections coming in to the monitor. What you didn't say is if ALL of the cameras do the same thing or if the side cameras are OK and it is just the rear camera that acts up.
The cables connected rom the cameras to the monitor are shielded and should not be getting any interference from the engine IF the system is properly grounded to the chassis. When checking the connectors, also make sure that the ground connection to the monitor is secure.
If just your rear camera is acting up, you may also want to check the connector for the rear camera. On the systems that I have had, the rear camera has a short pigtail that is connected to it and you have to pull it out of the rear cap to insure that it is plugged in to the long cable running to the front monitor.

Possibly loose connection, running engine causes interference. I would pull the monitor out of the dash and make sure all connections are snug.