We haven't yet stayed in or near Teignmouth, but we have wondered about whether the NT car park signposted "Little Haldon" on the B3192, about 3 miles out of Teignmouth, might be OK. We've driven past a few times and it looks a possibility, although a bit closer to the road than we would normally like.
I'd be surprised if there's any place for a stopover actually in Teignmouth.