I'm late to this (can't believe I missed a parking thread!), but have RTFT.
I'm generally on the side of OP, although she possibly did slightly over-egg it; then again, as a recent PP said, some people do think that they can do anything they like, then say sorry several times, and basically get away with it.
In fact, when all you do is keep saying 'sorry' a number of times, it can often reach a point where it morphs from meaning a (genuine or lip-service) apology initially into an implied PA 'you're being unreasonable and I'm just having to pacify your unreasonableness by repeating the same word until you realise and accept that you're the unreasonable one'.
Just saying the word itself means nothing if it's not said with any genuine sense of apologising. Most people would have held their hands up and said "I'm a carer, couldn't find anywhere to park and an elderly disabled gentleman was waiting for me to feed and change him, so I made a bad quick decision. I'm very sorry - I shouldn't have done it and I promise I won't ever do again." I know OP says he was young, but he was at the very least 17 if he was driving - a bit old to be looking to the ground, shuffling with his hands in his pockets and saying "Dunno, sorry".
He also outright LIED to OP by saying he 'didn't know' that it was a drive, when it must have been extremely obvious - as well as all the signs saying 'private property'.
He was doing an important job, but he had no idea whatsoever about the personal circumstances and job/daily schedule of OP and her neighbour. When he parked up at 7:30 in the morning, she could have been a hospital worker, soon to return from a night shift, and then heading straight to bed for a delayed 'night's' sleep in advance of another night shift. He isn't the only carer/key worker/emergency worker in the country.
That said, though, the worst CF in this scenario is getting away with it completely unblamed. I actually think that OP should consider contacting the care agency/council, but in a positive way. This man is doing an essential job, but is neither provided with the time nor the facilities to do it properly. If they can't ensure that he has somewhere close to the client to park (maybe having to carry in bulky supplies/food/medical equipment) - even having got him a parking permit on an oversubscribed road - the onus should be on them to pay him for the extra time it takes him to park as close as possible in an available, appropriate space and then go to the client's house. I believe Royal Mail take this into account, by giving a much shorter round and transport if necessary to their workers delivering to remote houses and those with very long drives, so goodness knows how a care agency is allowed to just blatantly ignore this need.