Depending on if you care about paying for parking, you can probably expect 20€ a day or so in a proper public garage. That said, parking is free outside of the M30 ring road.
I know there are a couple hotels near the Pinar de Chamartín metro stop, but that is probably 30-40 minute ride into the city center. If you stay at the Holiday Inn Pirámides you can drop your bags then drive to the other side of the river and find a spot for free and make the 10 minute or so walk back. Best time to arrive is probably around 5 PM as it avoids most traffic, but it's before parking spaces become a very precious commodity.
Also, keep in mind all parking is free after 3 PM in August (though there are neighborhoods with 24h restricted access in the historical center)