Are the parking lights on a 2003 Chevy venture supposed to stay at all times?

No, the parking lights on a 2003 Chevy Venture (or any vehicle) are not supposed to stay on all the time. They are only meant to be illuminated when the parking lights or headlights are activated by the driver. If they are staying on constantly, there's a problem in the vehicle's electrical system, likely a short circuit somewhere. You should have this checked by a mechanic immediately to prevent battery drain and potential fire hazards.