That depends. If the OP were transported from the scene and then issued the citation (either at the jail or the police department), more than likely, yes. If cited and released at the scene, then, no. In addition to the statutory authority (VC 22651(h)) a community caretaking exception must also be met. Since one cannot generally leave a car parked on private property, or if it was unsafe to leave the vehicle parked where it was, the vehicle can be impounded.
The OP also points out the vehicle's having dealer plates but being legally registered. That leads me to wonder about other issues - such as the lawful registration. Since january, newly purchased vehicles without metal plates must have a digital (paper) "plate" issued by the DMV through the dealership - NO dealer plates. If the vehicle was impounded in July with dealer plates (and by this I assume he means the dealer's name on a paper ad where the plates should be) then I suspect it was ether unregistered, or, displaying improper evidence of registration ... though, if the latter, I would expect another infraction or misdemeanor cite or booking for that offense (depending on the details).
Since we have no real idea why the vehicle was towed, it's all speculation at this point.