When one plant grows on another without harming it, it's an epiphyte. Tiny ferns and mosses growing on tree trunks are epiphytes.
In rainforests, orchids and bromeliads are common epiphytes, attached to other plants but getting their nutrients from the air around them. In more temperate regions, you might see lichen and moss growing on dead trees and guess they're responsible for killing their host — but these epiphytes simply take advantage of the increased sunlight and physical support of the bare trunks. You can grow an epiphyte of your own, too, by buying a tiny "air plant."