Who is responsible for compensation?

Hello
I recently flew from Manchester to NYC, the booking was with Thomas Cook. It was a BA flight operated by American Airlines. The booking codes all began with BA.
The flight was cancelled and we were delayed almost 5 hours while they had to get a replacement plane.
I've been chasing BA, however they say as it was operated by AA it is them who are responsible for the compensation.

Can anyone advise if this is correct.

Thank you