This is an English Heritage site so you can't take off or land within the site without their permission.
I parked in the EH car park (£3 for non-members) and walked to https://w3w.co/cardinal.hotels.satellite, which is on a public footpath. There was a handy tree stump to set up gear clear of the muddy path.
Accessibility: the footpath is accessed via kissing gates and a few steps, and can get very muddy in wet weather.
Hazards: the castle sits on a rocky outcrop roughly 120m high, so your drone ceiling may need adjusting to fly higher. However there were several low-flying light aircraft when I visited, and they may not all maintain full clearance above the castle. Also dogs and dog walkers.
Land owner permission not required.
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Co-ordinates: 53.12783, -2.691793 • what3words: ///quietest.soils.keys