We study the fixed parameter tractability of the restrictive $H$-coloring and the restrictive list $H$-coloring problems, introduced in~cite{DST01b}. The parameterizations are defined by fixing the number of pre-images of a subset $C$ of the vertices in $H$ through a partial weight assignment $(C,K)$. We define two families of partially weighted graphs: the emph{simple} and the emph{plain}. For the class of simple partially weighted graphs, we show the fixed parameter tractability of the list $(H,C,K)$-coloring problem. For the more general class of plain partially weighted graphs, we prove the fixed parameter tractability of the $(H,C,K)$-coloring problem.