No. It's a constitutional monarchy. The difference is that the the monarch still has some political powers even if it's mostly symbolic. In a real republic like the United States, France, Germany, etc, the parliament has all the political powers. The Republicans are trying to fool the people by saying that Obama is a monarch but they are wrong.
The political powers belong to the people in theory because many people argue that the judicial system, the media, the lobbies and other forces are taking powers away from the elected people but that is not in the constitution. Anyway, the judicial system applies the laws decided by the parliament not the other way around.