Car dealer using housing estate as overflow car park.

A vehicle only need's valid road tax to park on a public road.

It may be a local Authority issue but based on what OP is saying its not a case the Car Dealer is parking numerous cars at one location in the estate and instead parking individual cars at various locations. So it could be argued (I'm not a legal expert) the dealer has a right to park a car at a location on any estate unless a car parking permit is required (not something I've heard of in an estate setting) . Personally I can't see a local authority getting bothered by this.

I agree the Tax/ Insurance thing is mute.

If a car has not moved for 4 weeks, you can report it as abandoned.

But I would enquire with planning dept of the council as it's using public space for business use.

A council is quick enough to get a business to move tables off a footpath if no payment has been made.

Apologies, Typo 😉 Great user Name by the way Mr Wordle 😁

Does it cause difficulties getting in and out of your estate?

In our estate we have people parking at the entrance (who do not live in the estate), dangerously close to the junction when annoys the hell out of me. We also have people who park their cars on both sides of the road but won't use their driveways and want to use that space for other things. Sorry, my rant isn't very helpful, just that I feel your frustration.

Just to clarify it is a legal requirement to display motor tax, NCT and insurance discs.

The exemption for dealers not to display an insurance disc only applies where a trade licence is displayed on the vehicle.

You can fill out any mix of numbers when your filling this out.

I think you could practically input "Imaclown1" and your tax disc would be issued out to you. Plus what if your renewing your motor tax for the 1st of March and your insurance is up on the 20th of March. You would be inputting your valid policy number in that scenario but your motor insurance may not be renewed come 20th of March.

True but a vehicle can’t be towed away simply because it has no insurance or nct displayed

Can be towed and impounded for no insurance and tax out for more than two months.

It shouldn't come to that for the OP's case though they just need the dealer to stop parking in the estate.

When taxing a car online you can put any random number in as your insurance number and it will work.