ParulaCM = Module[{colorlist}, colorlist = Get["parula.txt"];
   Evaluate[Blend[RGBColor @@@ colorlist, #] &]];

mat = ImageData@
   ImageResize[ExampleData[{"TestImage", "Man"}], {256, 256}];
im1 = Colorize[Image@mat, ColorFunction -> "Rainbow"] // AbsoluteTiming
im2 = Colorize[Image@mat, ColorFunction -> ParulaCM] // AbsoluteTiming

(* "Rainbow": 0.063 seconds *)
(* ParulaCM: 1.267 seconds *)