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Struggling with two clues here
First "Where to park behind country mansion (4)". Have ?E?T but no idea of the answer.
Second "Airline after boarding puts off those who argue (8)". Have only D as first letter. Thought it was DISSENTER but that is 9 letters

Seat

A place to park your behind.
And a 'Seat' is also known as a country mansion where hierarchy would reside.

Debaters (those who argue)

BA (British Airways inside (boarding) deters (puts off)

An example.

https://en.wikipedia.org/wiki/Woburn_Abbey