Thanks.
The date of the parking event and date of issue are one of the things that can determine whether or not you are potentially liable as the registered keeper under Schedule 4 of the Protection of Freedoms Act 2012. In this case I can see that the parking event took place on 13th May, so they're within the relevant period.
Whilst you are wise to be cautious, I wouldn't worry about identifying yourself to ParkingEye for 2 reasons. Firstly, they already know who you are. Secondly, they issue roughly 2,000,000 parking charges every year (1 every 15 seconds or so) - they're unlikely to be trawling these forums to try and catch out the small few who come here for advice.
The signage you've shown us looks pretty shoddy and could form the basis of a defence. If you're able to, it would be sensible to get some more photos, showing the route the driver took to reach that parking area, and any signage on the way (or lack thereof!).
Outside of any appeal process, it's also always a good idea to speak to the PALS department at the hospital, to see if they will assist.