What type of Moss is this?
Mosses are part of the Phyla Bryophytum. They are self-contained plants, only dependent on other plants for structural support. They take up water and nutrients from the air and rain around them, not from the plants they live on. They are not parasitic in any way. The only harm mosses can do to plants is if they are so overwhelming that they weigh down the plant. This article, Living in the Land of Mosses has more information on dealing with mosses in your garden.
Identifying mosses at the species level requires more expertise than I have. Here are a couple of articles on types of moss, Bryophyta common mosses from the US Forest Service, and Mosses to know in the Moss Garden in the Bloedel Reserve in Seattle which may help you identify the type of moss. Living with Mosses from OSU has extensive information on mosses.