I don't think the BBC supports a political party, but support for certain positions is certainly evident. They are clearly pro-EU and anti-Israel, for instance.
I went to a "meet the senior managers" public feedback event at the BBC a few years ago. The Chairman, Director General, controller of BBC 1 and a few other top level people were there to hear the views of the public.
They were repeatedly, in turn, accused of being both pro-Israel/anti-Palestine and anti-Israel/pro-Palestine by various audience members, each using their own cherry-picked examples.
If you can attract that level of criticism from both sides of the argument, I think you can lay a reasonable claim to being fairly neutral. They certainly weren't "clearly" anti-Israel to some people.
