I've had a certain scene in my mind about your parrot featuring John Cleese.  I'm not a pet keeper or have anything other than a search engine. I found this link which looks useful for you. 

http://www.jamescargo.com/pets/pet-birds.php

There is general info here too. 

https://en.wikipedia.org/wiki/Grey_parrot

In October 2016, the Convention on the International Trade of Endangered Fauna and Flora (CITES) extended the highest level of protection to grey parrots by listing the species under appendix 1, which bans global and domestic trade in the species.[15]

It does not look easy as he/she may be Psittacus erithacus which is a protected species on CITES Appendix 1 

https://cites.org/eng/app/appendices.php

You will need to read on to understand the status. 

I guess you need to know about the breeding status/age etc of your bird. 

Consider the the risk of bird flu - it only takes a visiting sparrow to pass it onto your pet and on to birds you encounter on your travels - and returning to the British Isles. 

No idea if a vet inspection or quarantine apply. 

https://www.gov.uk/guidance/avian-influenza-bird-flu

https://www.gov.uk/guidance/export-and-import-licences-for-controlled-goods-and-trading-with-certain-countries#do-you-need-an-export-or-import-licence-for-animals-and-animal-products 

https://www.gov.uk/government/collections/guidance-on-importing-and-exporting-live-animals-or-animal-products

https://www.gov.uk/take-pet-abroad

https://www.gov.uk/bringing-food-animals-plants-into-uk

You will need to read all this information on each country website that you intend to visit. 

There are still different rules in other the EU countries as there are here. For example, my dad lives in Germany and was after a bike rack last time he visited. He would not buy a particular bike rack as it did not have a TÜV sticker on it, he said the cops could easily give you a ticket or at least a hard time. In Germany, there are also strict rules about what emissions your van has, and a sticker which restricts which towns/cities you can enter.   

In your case, you may risk surrendering your pet, so you will need to know the local law and language to effective argue your case for carrying a CITES protected species around in a campervan. 

Sounds like BIG a headache to me - besides, there are lots of stunning places in the four countries of the UK. As first timers, get to know your van at least.