First, a list of dates:
15 April 2002 in Northern Ireland
1 December 2003 in England and Wales
4 May 2006 in Scotland
If the child was born on, or after, the relevant date in that list, the fact that the father's name appears on the birth certificate automatically gives him parental rights. The child's 'official' name (i.e. the one that's used on his passport, and other official documents) can't be changed without the consent of everyone who has parental responsibility. However the child's guardians can use any name they like for 'everyday' purposes.
For an expanded answer, please see my post here:
http://www.theanswerbank.co.uk/Law/Question762 014.html
Chris