That website is owned by the Association of Train Operating Companies ('National Rail') and is specifically intended to promote travel on the National Rail network. Most Oyster pre-pay customers are using London Underground services, which are independent of National Rail, so it's unsurprising that you can't get the discounts with an pre-pay Oyster card.
If you live in outer London, it might be worth buying a ticket to central London (or wherever else the attraction is located), on a route which is operated by National Rail, rather than using your Oyster card.
http://www.daysoutguide.co.uk/faq.aspx#2
Chris