The freedom of speech is not simply a private that only those who you agree with can enjoy. It is a universal right shared by every citizen of a democracy or democratic republic. Every individual has the absolute right to free speech, even if they have wildly unpopular views. Do also consider that if you can silence your opposition's speech, the reverse can easily happen if your opposition then gains power.
They should be allowed to express what they think, as should everyone else. However, the other parties, or people, should be able to reply to those extremist views with the views they hold and why the extremist views are bad. Freedom of speech is important for all societies and political parties should be able to form and express views freely.
What is extreme, or not, is a relative question. In terms of immigration, much more of the British population may actually agree with the BNP than say, Labour, thus making a case that the Labour Party is actually 'extremist'. Regardless, the right of freedom of speech protects the airing of all views, even those deemed extreme by some.
I believe the British National Party should be allowed to air their extremist views. For me, I like it when parties tell on themselves by supporting extremist ideas because it is far easier to note the organization for what they really are. In the end, I think it is far better to have this honesty out in the open rather than hid behind closed doors.
Many people do not like the viewpoint of the British National Party. However this does not mean that they should be forbidden from airing their views. Freedom of speech is a key human right and the National Party and its members should be allowed to fully exercise that right. Unpopular speech should not be repressed merely because it is unpopular.
When people say whether they should be allowed, its confusing what they mean - after all, what are people advocating? Banning them from speaking of certain topics in certain ways? If the BNP is being elected to positions then they clearly represent some portion of the people in the country.