Sunday parking in Central London near Natural History Museum

Arfa__

I've always found this site good: http://www.parkopedia.co.uk/ Helpful to find the cheapest car parks nearby. No idea about free parking around there, sorry.

Pincher

A counter intuitive solution is to use a Car Club car during the week. Dedicated car bay, congestion charge inlcuded. About 50 miles free PER rental.

It only works if you work in central London, so you can pick up the car after work. Zipcar has overnight rates, 18:00 to 08:30 for £20 roughly, Monday to Thursday. Weekends do not offer overnight rates.

So, you pick up on Monday night at 18:00 after work, drive home,
use it for weekly shopping at TESCO.

On Tuesday morning, drive the family to central London, and RETURN the car, so you stop paying during the day.
Have your day out, no pressure of meter running. Alternatively, you go to work, wife takes kiddies during the day.

After 18:00, pick up the car, could be a different one, start using it for going to theatre and restaurants. Quite a lot of free parking after 18:30 in central London. At the end of the evening, drive home.

On Wednesday morning, drive in to central London, return the car by 08:30, go to work. By renting in two segments, you get 100 miles free.

I know it's not exactly what you were asking for,
but I think it's worth mentioning for M25 folk.