Can I drop a disabled passenger off at the castle side of the esplanade?

Which castle are you referring to?

Ha, I was wondering that myself. There are a LOT of castles in Scotland. :-) Although I do believe that Edinburgh Castle has an Esplanade, so that is probably what Clare is referring to. Here is an access guide to Edinburgh Castle that should give you the info you want: https://www.edinburghcastle.scot/plan-your-visit/access-guide

It would Edinburgh, The Esplanade is the area at the front of the castle.

There is no public parking at Edinburgh Castle. Castle Terrace and Johnston Terrace have the nearest on-street parking zones.
Castle Terrace NCP is the nearest car park.
There are a small number of parking spaces available for ticket holders with Blue Badges on the Castle Esplanade. These are available on a first-come, first-served basis.

https://www.edinburghcastle.scot/plan-your-visit/facilities

Your GPS should be able to give you directions. Be aware that there may be some road closures in August, or even in the preceding weeks in July, for the Tattoo and the Fringe. The Tattoo takes up the entirety of The Esplanade. It may be a bit of a faff to get to where you want to be, but it will be do-able in some manner.

ETA: The Blue Badge referred to is a government issued pass from a department called Mobility, which allows parking for those with mobility issues. I don't think a visitor would be eligible. On reading Mardee's link, it doesn't mention being restricted to Blue Badge holders there, and provides an email address. It may be a moot point in August anyway.

Gerry, you're right in that Blue Badge passes are not available for non-UK citizens. But there is a way for a disabled passenger to get up there without walking in the link I posted above (which may be what you are referring to).

Regardless, I think that Clare will have to contact the number/email listed to find out more information, and where exactly she can drop off the passenger.

The other thing is, if your passenger is disabled, are they able to tour the Castle?
Many steps, stairs and rough cobblestones.

But there is a way for a disabled passenger to get up there without walking in the link I posted above

If you're refering to this bit:

A mobility vehicle to Crown Square is available on request, and can be arranged by staff at the castle entrance. A maximum of three passengers plus one passenger that uses a wheelchair may use the car at once.

That's once you're inside the castle walls, a ride to the high parts up the really steep hill.

Dropping off at the gate might be a bit of a nightmare overall, but I assume the OP has good reason for taking their own car rather than a taxi.

Oh, you're right - I didn't notice that. Yeah, and what S J said above is true. I don't think that anyone who cannot walk up the Esplanade would be able to walk around the castle.

https://www.historicenvironment.scot/visit-a-place/places/edinburgh-castle/access/#:~:text=Visitors%20access%20the%20castle%20over,staff%20at%20the%20castle%20entrance.

Here is a bit more information.