In Hungary you must only park on the side of the road in the flow of traffic, therefore the right hand side. On one way streets you xcan park on both sides as long as cars can pass through.
Parking is generally metered on the street although the country has tried to help tourists by making parking free near the sights. Parking meters give you a short length of stay and to park for longer you'll need to use a municipal or private garage parking.
Enforcement of parking is done by the police and you'll either get a ticket, or if parked for too long, a clamp along with details on how to get your car released.
Acknowledgement of the terms of the European disability badge scheme isn't widespread in Hungary but it's worth taking it to show that you need extra assistance. It's best to use parking garages where the attendant will almost certainly help you gain an accessible space.