No. The approved housing development at Old Malling Farm, Lewes, is on what the South Downs Society calls "quality agricultural land". And it's been through a public inquiry, which ended with the Planning Inspector approving the planning application.
The National Park planners have just approved hundreds of new houses being built on the outskirts of Lewes, despite the objections of the usual NIMBYs. There is absolutely no evidence that the South Downs National Park Authority are against new development.