You want the good or the bad news first?
Well the good news is that you can park for free on streets, on holidays and during the weekends. That's considered to be friday 10:00pm until monday 9:00am.
(on all streets except for the main shopping street "Mariahilferstrasse"). Be careful not to block loadingzones, handicapable parking or gates that may have cars exiting. Loading zones are called "Lieferzone" in german, if you see that sign, dont stand there in between the hours that it shows on the sign. Also note that we are in the city center, so parking space is limited and may take you 10, 20, 30 minutes to find a spot. During Christmasmarket dont even start looking...
The bad news is that parking on the street during the week is limited to a maximum of 2 hours, so even if you are willing to pay for it, its not possible. Parking Tickets for the street cost 2.20 Euros per hour but as said, every 2 hours you would have to physically exchange them inside your car. Tickets can be bought in Tabak shops or inside the Metrostation from a machine.. (There are no machines on the street)
Another solution is the Public parking garage around the corner of the Hostel. The price is 33 EUR for 24hours (which is reasonable considering that street parking costs 2.20 EUR per hour)
We recommend searching for P+R (Park and ride) on google and park the car there, this is outside the city center and you will have to take the metro but it is by far the cheapest option coming to about 4 euros per day. (Prices may differ, please look it up online)