What a sad, boring life these people have if their only hobby is protecting the public-owned space in front of their own house - which then drags in all of the neighbours and makes their lives that bit more annoying.
I'd be interested to know if these selfish idiots who believe that only they are allowed to park on the road in front of their house extend that belief to others and therefore would categorically refuse to consider parking on the public road outside anybody else's house.
I'd just double-bluff them - play them at their own game - and tell them that, as it happens, the bit of road in front of their house is actually my land - as can be clearly seen on my deeds. This is much better done when other neighbours are there to witness it and share in the fun.
When idiot neighbour starts to froth and demands to see these deeds, tell them they'll need to get their deeds too, so that you can compare and decide the matter once and for all.
Whether or not either/both of you can show your deeds or you go as far as downloading digital copies, or even checking with the council, is neither here nor there. What is important is that you will have to exaggeratedly pretend to eat humble pie and concede that you do not own the piece of road, as you have no proof that it's your property - and exactly the same will be true for idiot neighbour as well.
What possible justification can they then have but to categorically concede that it must in fact be publicly-owned road and therefore equally fair game for anybody to legally park there?
All they're left with is to throw a tantrum and start whining pathetically in front of everybody "But I LIKE parking outside my own house!". In which case, just reply "Well I'd really like to share my street with reasonable neighbours who act like grown-ups and just let each other get on with their lives - but we can't always get what we want, can we?"
Bank their pathetic whiny toddler tantrum well in your memory so that, should they ever be stupid enough to approach you again, ready to complain, you can just perform it straight back to them until they go away fumming but soundly defeated.