I've had a (sort of) clarification from Richmond Park:
The route between Robin Hood Gate and Spankers Hill Wood is part of the Capital Ring cycling route. The route between Roehampton & Kingston Gate and Kingston & Richmond Gate is using the Tamsin Trail which is a shared path between cyclists and pedestrians.
Which is to say, they would prefer you not to use the roads, and go along the off-road route. However, on pressing, I got this:
Cyclists are allowed on the roads in Richmond Park the same as cars. On the map where there are already dashes indicating either the Tamsin Trail or the National Cycle Route we have not put the red dashes even though cyclists are allowed on them which is explained on the key symbols of the map.
Which is to say that we
are allowed. The bit about the Tamsin trail is misleading, as that doesn't use the road at any point, as far as I can tell. I'm not sure about the NCR, but I think it's actually along the Tamsin from Roehampton to Richmond Gates.
The Capital Ring, which does go (off-road) from Robin Hood Gate over Spankers Hill, and out of the Petersham pedestrian gate, is a walking route, and not really for cycling, unless my correspondent is advising that it is permitted just in RP (which I doubt). I don't think I'd risk the ire of pole-wielding walkers to make a pretty poor point on that one.
http://en.wikipedia.org/wiki/Capital_Ring
Of course, what has been overlooked in the email is that the road from Sheen Cross to Ham Cross is open to cyclists, and not motorists, but that's a minor niggle.