Okay, further research tells me that the first Robinmobile was an anonymous gift from a villain in the hopes, Robin would lead them to the Batcave, so that doesn't quite qualify as Dick being allowed to drive, while the second (seen in my previous post) was a Batmobile Dick retrofitted when Batman was missing. The latter would actually explain why Dick wasn't allowed to drive the Batmobile, as Batman probably wasn't too happy about Dick messing with his ride.