What's it like to live in Brixton / Brixton Borders

Talking to a couple of shop-keepers in and around Dulwich and Forest Hill the prevailing view is that shops with parking outside/nearby are more successful.

Yes, usually based on anecdotal evidence. Every shopping survey ever conducted in the inner city shows the opposite--car parking can actually serve to make an area less attractive to people on foot. Cars take up a lot of space that could be used otherwise. Quite a lot of time, the people parking outside (without moving their car all day) are the shopkeepers themselves. I could imagine that many of the market stallholders would have used Popes Road car park when they were at the market, though, and so its closure, unless there was other car parking for them nearby, would certainly have affected them. As many retail locations in Inner London don't have any off-street access, space for deliveries is very important, and market traders need somewhere to park if they're carrying heavy stuff. Many market traders don't; I was surprised when we did a lightning survey of Broadway Market traders recently and found that most of them came by bike/cargo bike--many weren't selling anything heavy. One stall had a lock-up nearby.

Even if this is not true (ref Olly's stats) the shop-keepers themselves will migrate to a location that has parking.

They will go to a location with higher footfall first. It will be well-known to them which locations are more lucrative. Such a location may, of course, include accessible car parking, but it's very unlikely that this would be the reason for the quality of the site. It'll be much more important factors like proximity to public transport/high housing density around/good footways/good mix of shops/etc., but see caveat about business acumen below.

In some ways the perception is the reality- rent hike, no parking- stay or move?

Most would opt to move in that situation, even if the shopper footfall (according to the stats) remained the same.

Oh, I agree that people may well act on what they believe about these things, but it's usually prudent to believe these better-researched findings. Many of them aren't very good business people, which is why they run small shops. This is not to denigrate them--there are personal qualities required for running small shops that good business people often don't possess. You'll find that many are very conservative (not as in small-c conservative or large-C Conservative, just conservative) and don't, in fact, (want to) move around very much. Quite often, they're also stuck where they are for one reason or another. As ever, there are many factors to consider.

I'm not sure that perception becomes the reality; most of the time I'd say the reality is cold, hard economics.