In message , purple pete
writes

Maybe the walking time could be increased to 10 to 15 min max.

I doubt that you will get that close. There are strictly enforced
special parking controls on event days covering an area of about one
mile around the stadium:

http://www.richmond.gov.uk/doc-parking_zone_r.pdf

OR
alternatively park the car at another station and arrive by train. But
which is the best service?

There is a large NCP car park in Drummond Place, next to Richmond
Station. Frequent trains to Twickenham, only a few minutes away, but
then you will have a good 10-minute walk to the stadium. The RFU run a
free shuttle bus between Richmond Station and the stadium for rugger
matches, but I don't know if there is anything similar for other types
of event.

I really wanted to avoid having to come out and get a train just to
reach the car if at all possible.

On-street parking near the stadium is very unlikely - it is in a densely
populated residential area. You could look for one of the temporary car
parks that often open in school grounds and the like - one is listed at:
http://www.twickenhamparking.co.uk/ (still nearly a mile away, though,
and I recommend pre-booking if it is open on the day you want).
--
Paul Terry