One thing you might consider is staying on the outskirts of Salzburg. My friend and I had a car and we stayed at Pension Ballwein, which was just outside Salzburg. It was so nice, with gorgeous views from our bedroom terrace, quiet and peaceful and very reasonably priced. Our room was charming with lots of kitschy touches that seemed very Austrian, and the beds had very fluffy comforters and pillows. But the best was the view. There were French doors that opened onto a balcony, which gave way to an incredible vista of the Alps.
The bus stop was right across the road, so we just took the bus to and from the city center (about a 10-15 minute ride) and left our car parked at the pension. This worked extremely well. https://www.pension-ballwein.at/?lang=en
We had a room with 2 beds there, but they also have family rooms and apartments. I would definitely recommend a stay there and would stay there again myself.