I always use the Indigo car park. Easy access and OK exit. Always space and better secure. The Club used to use the Hilton but that was before the Mayfair. Might look at that for day trips.
I agree about not using a car but with 92 year old in tow a car is the least hassle option. My rule for driving in London is drive at traffic speed, give no quarter, avoid eye contact, restrict your visual scan to front sector 😈