As Clive said, park in the Red Lion car park. There's an honesty box by the entrance if you want to contribute - I always chuck a quid in, but it's not compulsory.
Lacock and Claverton are about the only parts of the BA that do still get fished with any regularity, but it's nothing like it used to be in the 90s - I lost count of the number of times I turned up at Claverton back then at stupid o'clock in the morning during mid-week in early summer (before the schools broke up) to find it already busy. These days you can turn up to both on a Saturday or Sunday in clement weather and there's a reasonable chance you'll have the entire stretch to yourself.
I would say Lacock has far more fish and far more variety, but I think Claverton still produces the largest of most species, albeit very few and far between.
Lacock is still quite busy in all but the very harshest weather conditions all year round with non-anglers (abbey visitors, dog walkers, joggers, wild swimmers, picnickers etc etc) but it's still a lovely place to fish with a lot of accessible swims.