Hi Spanishfly,
Can't help on parking I'm afraid, but if you can get into Premier Inn County Hall, then definitely stay there. Really good sized family rooms, comfy beds and because its a big old building, they have solid thick walls! We enjoy the breakfast there, unlike other Premier Inns, a huge choice, eat loads and it really sets you up for the day. You could book it for the first day, see what you think and then decide whether you book it for the rest of your stay or go elsewhere. There is a small newsagent opposite the entrance, useful for water and snacks for the room or your trips out. (If it's fully booked, and no it doesn't take part in the offers, there is a new Premier Inn just behind it that is fine, we stayed there last time).
Great location as you will be close to the London Eye, book your timed slot before you go, also in the same building is the London Aquarium, its good, the sharks are fantastic, but it is expensive, I would only go if kids mega keen and I could find special offer somewhere.
You are on Southbank here, which is a lovely place to wander with street entertainers, musicians, skateboarders, you can see the boats on the Thames. Great choice of restaurants along here, Giraffe, Wagamamas in particular are always popular for us. If little legs fancy a good walk you can walk to the Tate Modern and have a look round there.
From under the London Eye you can get a boat that takes you to The Tower. Definitely recommend this, we had a great time there, great tour with the guard full of stories, amazing up close to Crown Jewels, decent food.
You can also pick up the open top bus ride from the bus stop near the London Eye bridge. If you go to a Tourist Info website, I think you can get the ticket and it gives you money off London Eye, money off boat etc. You can also pick up the usual london buses from here too, which will get you to most places. Or take the tube from Waterloo station which is situated just behind the Premier Inn. It's all signposted.
I wouldn't waste time at Madame Tussauds, mega queues, heaving, and SO expensive. Mine love Ripleys in Piccadilly Circus, there are usually offers around, i got really good prices last time through Tesco, but again I wouldn't pay full price to go. From there my two like to eat at Planet Hollywood (again look for vouchers!) good menu choice.
Its good to see the Changing of the Guard at Buckingham Palace, google the timings and info. You could then walk up the Mall to Trafalgar Square and see the big lions. The National Gallery is there to pop in and see some mega paintings. From here you can walk to Covent Garden, fun for entertainers, shops, market stalls. Good M&S outside the tube station for sandwiches/picnic/treats!
The Natural History is in South Kensington and has the dinosaurs and it is also joined to the Science Museum, so that is a complete day out!! Exhausting but really good. Take a packed lunch to save money and you can eat it downstairs where they send the school trips or outside on a bench. If sunny and DS's still have energy walk north to the Princess Diana park, and have a nice play before getting bus/tube back to hotel.
Hamleys is also worth a trip, it makes a good loo stop too! It's on Regent Street, which you can walk up to from Ripleys.
I'd do three nights if I were you, helped by using lots of tokens where you can find them, packed lunches and not rushing! And you'll have a fab time. (in the summer it will be busy 7 days a week).