Gandhi, Nelson Mandela, Lenin, Abraham Lincoln, Barack and Hilliary, 25 out of 42 U.S. Presidents, 2 out of the last 5 Prime Ministers... all former lawyers. Since the 2010 general election, around 80 MPs are lawyers or have a law degree.
The skillset of a lawyer should lend itself to making a good parliamantarian: The ability to handle a brief at short notice, conduct meetings and effect good communication through presentation and advocacy, acting as a representative, understanding legislation and the legislative process, knowledge of the institutions of the state and its constitution, moulding solutions out of knotty issues, a public service ethic.
For practical ways of gaining experience and contacts at entry level, see www.policyjobs.net and www.politicaljobs.net, which require a subscription but advertise internships, analyst, research and campaign positions throughout the world.
UK specific, www.internsnetwork.org.uk does exactly what it says, and although not politics specific, has some useful guides.
Then there is 'Work for an MP': www.w4mp.org with listings and guides.
Contacting party political headquarters directly would reveal any of their available opportunities (see www.gov.uk), and also local constituency offices. Also see of course 'Civil Service' on the adjacent list, and the contacts thereon, e.g. Fast Stream.
For opportunities in government-funded public bodies (quangos) see
See www.lgjobs.com for information on careers in local government and council vacancies.
Also, see 'Local Authority Lawyer'.