While looking this up for someone who didn't believe it, I found that it appears no longer to be true.
In The Royal Parks and Other Open Spaces (Amendment) etc. Regulations 2010, S1 (2) an important change was made, namely:
1.—(1) These Regulations may be cited as The Royal Parks and Other Open Spaces (Amendment) etc. Regulations 2010 and shall come into force for the purposes of regulations 1 to 4 on 6th April 2010 and for all other purposes on 1st October 2010.
(2) In these Regulations–
“parking permit” means a permit issued by or on behalf of the Secretary of State for the parking of a vehicle;
“parking place” means a place shown on a notice exhibited by or on behalf of the Secretary of State as being appointed under these Regulations as a place where a person may park a vehicle; and
“vehicle” means a mechanically propelled vehicle intended or adapted for use on a road.
Note the parts I've highlighted in green.
Prior to that amendment, "vehicle" was undefined, giving it the usual all inclusive meaning.
The change to "mechanically propelled" excludes pedal cycles.
I'd guess that as the reduction from 30 to 20 mph was made in the previous amendment, it was not realised that this would start catching cyclists, and this was unintended, so in this latest amendment, this situation was rectified.
The original legislation The Royal Parks and Other Open Spaces Regulations 1997 had 30mph for Richmond. (Part II S1)
This was reduced in the Royal Parks and Other Open Spaces (Amendment) Regulations 2004 to 20mph in S2 (5).
My guess is that this was never intended to catch cyclists, and that situation has been remedied in the 2010 amendment - which brings it into line with other UK speed limit legislation.
Does that all make sense? (or as much as the law ever does!).