If you decide to drive to Budapest, you have the option of parking in the official airport car park from 6 590 huf for a week in the uncovered car park about 10 minutes walk to the terminal or 12 000 huf for a week in the covered garage right next to the terminal.
The prices of the official airport car park are very reasonable, the surrounding private car parks offer the same or even higher prices.
If you would still like to park completely free of charge in the unattended but official P+R car park, you have the option at Kőbánya-Kispest metro station, from where you can quickly get to the airport by bus 200E.