AIBU to think I can get accommodation & parking in London for £500?

I would park somewhere and get the train in. Premier Inn for the hotel, there are some nice ones.

Definitely get the train

There are Premier Inns near Paddington Station or in Paddington basin for between £300 and £400 for three nights for a family room. My understanding is that trains from Wales come in at Paddington so that might be an easy option and you will be near the tube for commuting. Do not drive.

We got an Airbnb near Clapham Common a while back that was £96 a night and paid daily rate to park at the Asda, which was £20 ish a day.

You’d be much better off parking outside London and train in to your hotel, leaving your car in any train station car park. Basically look at your route from Wales, find a station near London, look up parking facilities and cost of daily parking, choose one. Probably £20 ish a day, you can book multiple days at a time usually on whatever app the car park uses.

Get a family railcard and your rail fare will be even cheaper.
I never drive to London (from Yorkshire).

Any hotel in Ealing would work and only 15 mins into central on the Lizzie line. You’d have to cost travel in London but quite henestly I’d get the train and stay centrally.

The travel lodge and premier inn in Richmond will probably be within budget. Neither is a lovely hotel but both are ok for somewhere to put your head down. Richmond is on the District line so good for NHM, and train to Waterloo is 20 minutes. It's also a pretty pleasant place to stay, and you can walk by the river if the weather is ok.

The hotels might have parking?

Factor travel costs to central London time and money if you stay outside
Look for zone 2 premier inn like archway etc

3 nights is pushing it unless you can find a premier inn on offer, may have to be further out. I do drive to London now and park at Q Park Hyde Park which is around £60 a night but great location and frankly the trains are so unreliable every time we have planned to go there's been a strike or whatever so this suits us better. We normally stay at Holiday inn Bloomsbury but that's c. £200 a night.
If you stay on the outskirts hopefully you can find something:

Why not try train to Paddington (book in advance and travel off peak , much cheaper) , and stay there ? There are a couple of hotels in the area, but easy to use the tube to get as and where needed. Paddington has trains toward Cardiff and the South West which may make things easier. This one here is a bit more than anticipated , but admittedly has its own kitchen and can recommend it for staying and a good base. Paddington is great to wonder and enjoy , white not being directly in the centre.

It can be pretty expensive getting the train in - I live in Woking, about 30-40 mins in on the train to Waterloo, and weekday return tickets for the three of them would be £45 per day. Could get it down to £35 daily if you have a family and friends railcard. But if you wanted to travel before 9.30am it would be more like £55 daily. So be careful about train costs going in daily! And with a 10 year old I reckon it would be nicer to stay centrally so you can go out and eat in the evening then be back at the hotel quickly rather than heading out on a 45 mins journey back to the hotel each night.

You and your husband can use your debit cards on the tube, buses, trains in London. Just remember to use the same card. Your daughter may need one of these to get free travel https://tfl.gov.uk/fares/free-and-discounted-travel/5-10-zip-oyster-photocard#on-this-page-0

if you're parking outside of London and travelling in, why near Heathrow which is likely to be one of the most expensive options?
as a PP has suggested, if you're far enough out, free on street parking is an option - or an NCP,,, or the park on my drive parking apps - but I don't get why you'd go for somewhere premium like Heathrow.

Also, in some areas, your train journey into London could be combined with a travelcard, and some London attractions offer 2 for 1 type deals if you've travelled by train.

I would drive to Maidenhead and hop on the Elizabeth line there. I've done this from Wales. We parked on a driveway booked via "Your Parking Space" for a reasonable price, we had an easy ten minute walk to the train station and saved so much money.

dont stay in Slough or near Heathrow, grim areas.

If you’re looking outside of London, why not try Windsor as just off the M4 and good for heading back and having meals there in the evenings? About 50 mins into London on either the Elizabeth Line via Slough or SWR to Waterloo. Holiday Inn is £100 a night.

There is a large DoubleTree hotel with parking on Ealing Common - 5 min walk to either Ealing Common station (Piccadilly/district lines) or Ealing Broadway station (Elizabeth/District/Central Line). Also quite a nice area - 15 min drive to Westfield too.

We park at Westfield shopping centre (not sure about over 24 hour parking) or for free a short walk from Hanger Lane.

Yes I think you can do it but you're better off staying on the outskirts and getting the train in.
Have a look at Watford - we stayed at the holiday in express near the station when we went to HP studios and then got the train in to Euston the day after, very easy. We didn't have a car with us so don't know about parking.
It was about £70 for a night without parking.

Somewhere like Slough/St Albans/Windsor could work too, or Heathrow like you say.

I got a premier inn Paddington for £110 a night. Park at a tube station with parking £18 a day charge. Tube in. Totally doable

Look in the Kew area, I found a hotel for around £100 a few months back and parking (at hotel) was £15 per day. Make sure it's not too far from the tube

Actually it looks very doable for accommodation. Probably helps that it’s the last Sunday of half term and then a few days into school time (at least, my kids schools in Surrey are on holiday 17-23 Feb)

eta I posted a screenshot of premier inn search for your dates in London with prices etc on a map - guessing all images are being checked before publication at the moment.

We've been during school holidays and stayed at Covent garden and Victoria travel lodges cheaply enough. I think it may be quieter during the holidays becayse fewer business's people staying away.

Driving isn't a good idea, the ulez and congestion charges are expensive for most cars.

Not Heathrow, there's a huge premium for parking near the airport. But use JustPark to find a parking space near an Elizabeth Line stop and use vrbo.com to find accommodation - plenty of nice flats available for around £390 for your dates.

Yes this is a good location. Family room for the 3 of you on the nights you’re looking for OP is £265-£344 depending on your flex options. So plenty left for paying for parking and train a bit further out.

https://www.premierinn.com/gb/en/hotels/england/berkshire/slough/slough.html?ARRdd=23&ARRmm=02&ARRyyyy=2025&NIGHTS=3&ROOMS=1&ADULT1=2&CHILD1=1&COT1=0&INTTYP1=FAM&BRAND=PI

£169 for 3 nights

Shepherds Bush? Right there for Westfield, easy to get to Central (multiple buses/tube options) and easy access from the M40.

I wouldn't recommend Heathrow. It's extortionate both for parking and for then getting in/out of central (unless you take the long winding routes)