Geography : Parth Galen

[The Mouth of Sauron]

Would I be correct in thinking that Parth Galen was within the borders of Rohan ?

[Inziladun]

Technically, I think it was within the boundaries of Gondor. Some time after the Three Hunters left Parth Galen, we have this quote:

The ridge upon which the companions stood went down steeply before their feet. Below it twenty fathoms or more, there was a wide and rugged shelf which ended suddenly in the brink of a sheer cliff: the East Wall of Rohan.
TT p 15 (paperback)

[The Might]

Nope, as far as I know the eastern border of Rohan in that region was called the East Wall of Rohan, represented by the western ridges of the Emyn Muil, an impassable wall of rock. So everything east of it would theoretically belong to Gondor, including Parth Galen.

However, at the time of the War of the Ring it did not practically belong to Gondor as no Gondorians ventured out there. But I believe that it was part of the Reunited Kingdom later.

EDIT: xed with Inziladun.