When I was in St. Andrews a couple of years back, we usually parked in the small free carpark called Murray Place. If that doesn't work, you can look along the Scores, running up from the Old Course towards the University, there's a lot of free parking there. As others have said, once you've parked, its a short walk to anywhere in the town.
If coming in on the A91 you come to a roundabout with the golf course on the left and (I think) the university on the right. There is a large free car park there.
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