I feel they should answer to Parliament because they benefit from and contribute to Britain's reputation in a way an airline with any other name doesn't.
I agree. BA is one of those companies which nowadays operates on a "Now I'm one thing, now I'm not" approach. They are very happy to trade with the national name, to receive the lion's share of the grandfathered slots at Heathrow, to lobby for all that UK government passenger business, and so on. But they then take all the money and spend it in Madrid or Mumbai, say because they are an international business they shouldn't take part in anything from the UK Parliament or elsewhere that they find inconvenient, and such like.