Motorized bicycles, Mopeds and Motor Scooters are not the same things, and there are things that you can and can't do with one or the other, depending on your jurisdiction.
In VA, where my vehicles are registered,a 49cc scooter isn't a motorized bicycle, it's classified as a Moped, which can receive a Moped license plate, allowing for low speed operation, on public roads, without a driver's license. You could get a Disabled license plate, but it would have to be registered as a motorcycle, which comes with more registration and licensing requirements.
Generally, since you can't ride a motor scooter on the sidewalk, you can't park on it either.
That said, you can park a motorcycle or scooter a lot of places that aren't full-sized parking spaces; as long as they're not vehicle traffic or foot access.
Adding in the need to lock it to something is a consideration, and while it's usually acceptable to lock a bike to a convenient lamp post or handrail, it's usually discouraged to put your scooter on a bike rack