On private property, a driveway or a hard pad is the normal spot and that is where a container should go in Parma Heights wherever possible. The paperwork that does apply here is mostly about the work rather than about the container.
The Building Department is the office that matters. It handles building permits, contractor registration, business licensing, rental registration, inspections and code compliance for the city, and it is clear that contractors working in Parma Heights have to register with the City before they start. It also doubles the permit cost where work begins or finishes without one, so the sequence is worth getting right. The Department of Public Service handles business licensing and permits as well, so for an unusual question it is worth asking which of the two owns it. The City is at 6281 Pearl Road, and the Building Department takes calls on 440-884-9607.
On the container itself, the honest position is this. Parma Heights publishes its codified ordinances through an outside legal publisher whose site is closed to automated access, so nothing is stated here about a city container approval, a duration limit, a placement standard or a hauler license in either direction. If your container has to sit on a city street rather than your drive, call the Building Department before you fix a delivery date.
The county layer is clearer, and it is good news. The Cuyahoga County Board of Health runs the solid waste program for this county under the state rules, and what it licenses and inspects are facilities: landfills, transfer stations, construction and demolition debris sites, compost sites, scrap tire operations and infectious waste plants. It does not register solid waste collection trucks or license haulers. The only hauler program the Board of Health runs is for sewage, which is a different set of rules entirely. So in Parma Heights, unlike parts of the Dayton region, there is no county truck registration sitting behind the city rules.
Your Prime Dumpster account manager will confirm the arrangement for your address before the job is booked, and will keep the container on your own ground wherever it physically fits.