In this thesis we consider a globally defined holomorphic function,fwith a parabolicfixed point, and study the set of holomorphic germs which fix the parabolic fixed pointand commute with the given function. This set is called the local centraliser.In Section 3 it is shown that the holomorphic germs must respect the dynamics of thefand all possible group structures of the local centraliser are given. Section 4 focuses onthe case wherefis a polynomial of two terms and uses the parabolic case to prove similarresults for a generic two term polynomial with an elliptic fixed point.Section 5 and 6 study extensions of a given holomorphic germ into the immediate parabolicbasins off, provided that the parabolic basins are proper (see definition 2.28). Section 7then extends a holomorphic germ from the immediate parabolic basins onto a neighbour-hood of the closure of the immediate parabolic basins.Section 8 restricts to polynomials with connected Julia sets and studies extensions ofholomorphic germs onto the whole complex plane, under conditions of the ratio of criticalpoints outside the immediate parabolic basins offcompared to inside the immediateparabolic basins off. This is used to identify the local Centraliser of any polynomial ofdegree 2 and 3 with a parabolic fixed point and any polynomial of degree 4 and 5 with aparabolic fixed point and a connected Julia set.Section 9 studies quasi-hyperbolic polynomials. We identify the local centraliser whenthe interior of the filled-in Julia set offis a single parabolic basin.