My Park Avenue leveling system runs a lot after engine is shut off. Can a bad battery cause it to continue to run? It runs battery down in just a few minutes. Car will start immediately when jumped.

This air ride/leveling system is prone to leakage, causing the pump to run excessively, and thus drawing the battery down. The battery is not the fault but the victim.

There are several things that could be the root cause:
Either the height sensor isn't giving the correct height reading,
Or the system exhaust solenoid is leaking, or the air lines/air bags are leaking.
Likely the air bags have become dry-rotted and are seeping. As the vehicle slowly settles the pump will continue to operate.

My Park Avenue leveling system runs a lot after engine is shut off. Can a bad battery cause it to continue to run? It runs battery down in just a few minutes. Car will start immediately when jumped.
Most likely one of the bellow or air struts has a leak. However if your battery is getting drawn down in a few minutes, it is bad! I would replace the battery first; make sure your cars charging system is putting out at least 14 volts when running.