Does parliament select the UK prime minister?

No, since as Parliament is made up of opposing political parties, it would be impossible for a Prime Ministerial candidate to be agreed upon that would be acceptable to everyone!

The Prime Minister of the UK is the leader of his political party, achieving this position by rising up through the ranks of the Cabinet (or Shadow Cabinet) until his or her position as Party Leader is voted upon by the party. If they are leader of the Opposition, but their party wins the following General Election, then they become Prime Minister (as, for example, David Milliband will if the Conservative / Liberal Democrat coalition lose the next election in 2014).

If the leader of the governing party resigns as Prime Minister (as Margaret Thatcher did in 1990), then they are either replaced by their Chancellor (which is the second most powerful job in Government), or a new leader CAN be chosen by party ballot. For example, following Thatcher's resignation, John Major was elected as new Tory Party leader in favour of two other running mates, Michael Hestletine and another candidate I can't remember right now. But this is not usual practice- normally, if a PM resigns mid-term, then their Chancellor takes over until the next General Election (as Alec Douglas-Home did when Harold MacMillan resigned in '63, James Callaghan did when Harold Wilson resigned in '76, David Lloyd George did when Asquith resigned in '16, and so on).

The position of Deputy Prime Minister is subordinate to that of Chancellor- the Deputy PM represents the Prime Minister when the Premier is out of the country or on leave for some reason (i.e. illness, bereavement etc.), but does NOT take over if the Premier quits altogether. For instance, Liberal Democrat leader Nick Clegg is currently deputy PM, but if David Cameron had to leave office for some reason then it would be his Chancellor, George Osbourne, who would take over as Prime Minister, and not Clegg.